From Wikipedia, the free encyclopedia
.^ Prime numbers are numbers that only have two factors: 1 and itself. Tag: Prime  Explore content tagged Prime on eHow.com 28 January 2010 1:41 UTC www.ehow.com [Source type: General]
^ N is a NEW prime number itself OR b. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ Prime numbers used to be a mathematical curiosity.
The first twentyfive prime numbers are:
 .^ N=2x3x5x7x11x13x17 + 1 This number is not a multiple of 2, 3, 5, 7, 11, 13 or 17 because performing division you would always have a remainder of one.
 Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ First ten: 1 , 7 , 10 , 13 , 19 , 23 , 28 , 31 , 32 , 44 . Number Gossip: List of Properties 28 January 2010 1:41 UTC www.numbergossip.com [Source type: Reference]
^ The result is 31, 7, 19, 13, 21  or "VENIO", our original message. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^{[1]}
.^ So, for example, 1 in 6 numbers around 1,000 are prime. The prime number lottery 28 January 2010 1:41 UTC plus.maths.org [Source type: FILTERED WITH BAYES]
^ For each prime number p > 3, there exists a natural number n such that p = 6 n ± 1.
^ Euclid's second theorem demonstrated that there are an infinite number of primes. Prime Number  from Wolfram MathWorld 18 September 2009 15:41 UTC mathworld.wolfram.com [Source type: Academic]
.^ The reason for not allowing 1 as prime is to keep the fundamental theorem of arithmetic . prime number@Everything2.com 28 January 2010 1:41 UTC www.everything2.com [Source type: FILTERED WITH BAYES]
^ Prime numbers  A complete course in arithmetic . Prime numbers  A complete course in arithmetic 28 January 2010 1:41 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]
^ Representing natural numbers as products of primes 2 How many prime numbers are there?
Moreover, this factorization is unique except for a possible reordering of the factors.
.^ The property of being a prime is called primality .
^ The property of being a prime is called primality , and the word prime is also used as an adjective. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Proving a number is prime is not done (for large numbers) by trial division. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Our new number q equals the product of all primes in the set {2 ... Prime Numbers 18 September 2009 15:41 UTC www.arachnoid.com [Source type: FILTERED WITH BAYES]
.^ Does what you do allow calculating a big prime faster than existing algorithms? Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ In 200 BC, Eratosthanes devised an algorithm for calculating primes called the Sieve of Eratosthanes . prime number@Everything2.com 28 January 2010 1:41 UTC www.everything2.com [Source type: FILTERED WITH BAYES]
^ Extremely large prime numbers (that is, greater than 10 100 ) are used in several public key cryptography algorithms.
.^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Prime numbers, to me, are not all that interesting. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ All numbers are prime. Prime Magic Squares 28 January 2010 1:41 UTC recmath.com [Source type: Academic]
.^ A very significant one is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists: when looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows welldefined laws. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ He asked: What proportion of numbers are prime numbers? The prime number lottery 28 January 2010 1:41 UTC plus.maths.org [Source type: FILTERED WITH BAYES]
^ The first 30 prime numbers are: . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The results proof that the prime numbers are not uniform randomly distributed among the natural numbers. Prime Numbers Random? 28 January 2010 1:41 UTC members.tele2.nl [Source type: Academic]
This statement has been proven since the end of the 19th century.
.^ A very significant one is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The most significant of these is the Riemann hypothesis , which essentially says that the primes are as regularly distributed as possible.
^ It is trying to explain the sequence of prime numbers that the Riemann Hypothesis is all about. The prime number lottery 28 January 2010 1:41 UTC plus.maths.org [Source type: FILTERED WITH BAYES]
.^ There are many open questions about prime numbers.
^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ So, for example, 1 in 6 numbers around 1,000 are prime. The prime number lottery 28 January 2010 1:41 UTC plus.maths.org [Source type: FILTERED WITH BAYES]
.^ Problem 2: How many prime numbers are there? Prime Numbers  Dev Shed 28 January 2010 1:41 UTC forums.devshed.com [Source type: General]
^ The rate of convergence to infinity if there are infinitely many twins. Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
^ Goldbach's conjecture : Can every even integer greater than 2 be written as a sum of two primes?
.^ The notion of prime number has been generalized in many different branches of mathematics. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ K. Matthews, Generating prime numbers . id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ In ring theory, one generally replaces the notion of number with that of ideal. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Example using prime number 7. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ If it is a prime number, then this information is simply printed. Prime Numbers  Dev Shed 28 January 2010 1:41 UTC forums.devshed.com [Source type: General]
.^ Some special types of primes .
^ Special types of primes from formulas for primes . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The newly discovered number is an example of a specific type of prime number called a Mersenne prime, named after the 17th century French scholar Marin Mersenne. Why 2 to the power of 43,112,609  1 = $100,000 for prime number hunters  Science  The Guardian 28 January 2010 1:41 UTC www.guardian.co.uk [Source type: News]
.^ The largest known factorial prime is 3610!
^ Ten largest known primes . prime number@Everything2.com 28 January 2010 1:41 UTC www.everything2.com [Source type: FILTERED WITH BAYES]
^ Prime numbers p where 2 p + 1 is also prime are known as Sophie Germain primes .
^{[3]}
Prime numbers and the fundamental theorem of arithmetic
.^ A number n is prime if and only if it has exactly two positive divisors. id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ In mathematics , a prime number (or a prime ) is a natural number which has exactly two distinct natural number divisors: 1 and itself. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ A natural number that is greater than one and is not a prime is called a composite number .
.^ Prime numbers less than 2^18 . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Prime numbers are opposite to composite numbers. Number Gossip: List of Properties 28 January 2010 1:41 UTC www.numbergossip.com [Source type: Reference]
^ D. J. Bernstein, Distinguishing prime numbers from composite numbers . id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
.^ A prime has exactly one proper positive divisor, 1. id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ Or for short: A prime number is a natural number with exactly two natural divisors.
^ A number n is prime if and only if it has exactly two positive divisors. id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
Next, 4, is composite, since it has 3 divisors: 1, 2, and 4.
Using symbols, a number n > 1 is prime if it cannot be written as a product of two factors a and b, both of which are larger than 1:
 n = a · b.
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2 n ( Bertrand's postulate ).
^ Positive integers other than 1 which are not prime are called composite numbers . Prime Number  from Wolfram MathWorld 18 September 2009 15:41 UTC mathworld.wolfram.com [Source type: Academic]
.^ Primes are thus the "basic building blocks" of the natural numbers (The proof of this is below).
^ They realised that the primes are the building blocks of all numbers. The prime number lottery 28 January 2010 1:41 UTC plus.maths.org [Source type: FILTERED WITH BAYES]
^ Representing natural numbers as products of primes .
For example, we can write:

23244 
= 2 · 2 · 3 · 13 · 149 

= 2^{2} · 3 · 13 · 149. (2^{2} denotes the square or second power of 2.) 
.^ The same prime may occur multiple times. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ In the previous articles, the prime trial factors were taken from the same bit array as the numbers being tested.
^ The definition of the function may be to test a single number for primeness, but it may be called many times for many numbers. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
A decomposition:
 n = p_{1} · p_{2} · ... · p_{t}
of a number
.^ What about the 200 trillion you could run in a year  how many prime factors would you need?
^ It follows that s (N) is divisible by 2 k+1 1, which is odd, so this must be a prime (else it would factor into two odd primes). Geometry.Net  Theorems_And_Conjectures: Perfect And Prime Numbers 18 September 2009 15:41 UTC www.geometry.net [Source type: Reference]
^ Many prime factorization algorithms have been devised for determining the prime factors of a given integer , a process known as factorization or prime factorization. Prime Number  from Wolfram MathWorld 18 September 2009 15:41 UTC mathworld.wolfram.com [Source type: Academic]
to
p_{t} is called
prime factorization of
n.
.^ Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The Fundamental Theorem of Arithmetic states that for every number, there is exactly one way to factor that number into primes  and vice versa: every selection of primes multiplies into a different number. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ There are many open questions about prime numbers. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The set of primes is sometimes denoted , represented in Mathematica as Primes . Prime Number  from Wolfram MathWorld 18 September 2009 15:41 UTC mathworld.wolfram.com [Source type: Academic]
^ Because all nonprime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ However, since 2 is the only even prime (which, ironically, in some sense makes it the "oddest" prime), it is also somewhat special, and the set of all primes excluding 2 is therefore called the " odd primes ." Prime Number  from Wolfram MathWorld 18 September 2009 15:41 UTC mathworld.wolfram.com [Source type: Academic]
Examples and first properties
Illustration showing that 11 is a prime number while 12 is not.
The only even prime number is 2, since any larger even number is divisible by 2. Therefore, the term odd prime refers to any prime number greater than 2.
The image at the right shows a graphical way to show that 12 is not prime.
.^ Division by any other number less than the prime number results in a remainder. Testing For Prime Numbers 28 January 2010 1:41 UTC cpearson.com [Source type: Reference]
^ Twin primes : All twin primes except (3, 5) are of the form . Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
^ I said that all primes of are such form, not that all of such form are primes. Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
.^ Sum of k primes = Product of k integers . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ If p is a prime number and p divides a product ab of integers, then p divides a or p divides b . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ One of the central thereoms of Number Theory provides that the density of primes is always greater than a known number for the first incomprehensively large number of integers. Testing For Prime Numbers 28 January 2010 1:41 UTC cpearson.com [Source type: Reference]
.^ This proposition was proved by Euclid and is known as Euclid's lemma . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ It is used in some proofs of the uniqueness of prime factorizations. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Already some people are using keys that, in order to factor with the Number Field Sieve, would require more energy than exists in the known universe. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Why was the process so significantly slower using prime factors up to a billion?
Primality of one
.^ The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Postulate 1 essentially defines the primes...and does so by elimination of all the other numbers, which are composites...composed of components other than or in addition to one and themselves. "Butterfly Prime Determinate Number Array (DNA)" copyright © 2006, Reginald Brooks. All rights reserved. 28 January 2010 1:41 UTC www.brooksdesignps.net [Source type: Reference]
If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications, since 3 could then be decomposed in different ways
 3 = 1 · 3 and 3 = 1 · 1 · 1 · 3 = 1^{3} · 3.
.^ N is a NEW prime number itself OR b. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ For example, 7 is a prime number because it is evenly divisible by only 1 and 7. Testing For Prime Numbers 28 January 2010 1:41 UTC cpearson.com [Source type: Reference]
^ In mathematics , a prime number (or a prime ) is a natural number which has exactly two distinct natural number divisors: 1 and itself. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime, such as the work of Stern and Zeisel. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ It is not known if there are an infinite number of such primes (Wells 1986, p. Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
^ Main article: public key cryptography Several publickey cryptography algorithms, such as RSA, are based on large prime numbers (for example with 512 bits). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The Electronic Frontier Foundation (EFF) has offered a US$100,000 prize to the first discoverers of a prime with at least 10 million digits. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ HELLO! We've just printed the primes up to 10,000,000 in 2.30 seconds.
^ For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens? How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^{[5]} .^ Henri Lebesgue is said to be the last professional mathematician to call 1 prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^{[6]} .^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Numbers with exactly one prime divisor. id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ Properties of numbers that have a Mersenne Number as a factor . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^{[9]}
History
.^ Finding prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Prime number algorithm in C . Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ Find all prime numbers which is <= sqrt(lim) 2. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
.^ A more complicated, but more efficient algorithm (when properly optimized) is the sieve of Atkin. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ As a hint, there are faster but more complicated algorithms, like the Sieve of Atkin and the various Wheel Sieves. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
The Sieve of Eratosthenes was created in the 3rd century BC by
Eratosthenes, an
ancient Greek mathematician.
.^ D. J. Bernstein, Distinguishing prime numbers from composite numbers . id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ Prime Numbers  There is a pattern! Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ Therefore, if one is to calculate a prime – they have to know the number of “skips” that occur within the lines prior to the prime. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
.^ However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ However there is no evidence to suggest that starfish have 5 arms because 5 is a prime number. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ If p is prime and G is a group with p n elements, then G contains an element of order p . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ How about the next challenge is to return all 78498 prime numbers between 1 and 1000000?” . SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
^ Euler's Totient Function is denoted by the Greek letter phi, and is defined as follows: phi(N) = how many numbers between 1 and N  1 which are relatively prime to N. Thus: . Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ We already know how to exlude numbers from the prime list. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
.^ The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ F. Richman, Generating primes by the sieve of Eratosthenes . id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ So I decided to try to convert the Sieve of Eratosthenes to TSQL. This algorithm is known to be both simple and fast for getting a list of prime numbers. SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
.^ After the Greeks, little happened with the study of prime numbers until the 17th century. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime, such as the work of Stern and Zeisel. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Having the answer falling on Ray1 will ALWAYS happen no matter which two prime numbers you choose ... Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
.^ In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibnitz and Euler ). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ This can be deduced directly from Fermat's little theorem. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ A special case of Fermat's theorem may have been known much earlier by the Chinese. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ They include special cases such as the LucasLehmer test for Mersenne primes and Pepin's Test for Fermat primes. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Many wellknown conjectures are special cases of the broad Schinzel's hypothesis H. Many believe there are infinitely many Fibonacci primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Primes of the form 2 p − 1, where p is a prime number, are known as Mersenne primes, while primes of the form are known as Fermat primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Prime numbers, to me, are not all that interesting. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ They are called Mersenne primes in his honor. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Primes of the form 2 p − 1, where p is a prime number, are known as Mersenne primes, while primes of the form are known as Fermat primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ No shared digits between composites and its prime factors . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ However, the very next Fermat number 2 32 +1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The French monk Marin Mersenne looked at primes of the form 2 p  1, with p a prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Primes of the form 2 p − 1, where p is a prime number, are known as Mersenne primes, while primes of the form are known as Fermat primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ In 1747 he showed that the even perfect numbers are precisely the integers of the form 2 p 1 (2 p 1) where the second factor is a Mersenne prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ They are called Mersenne primes in his honor. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ They include special cases such as the LucasLehmer test for Mersenne primes and Pepin's Test for Fermat primes. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Fermat conjectured that all numbers of the form 2 2 n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ There are many open questions about prime numbers. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Prime numbers are of utmost importance in number theory. Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
.^ Adding the reciprocals of all primes together results in a divergent infinite series (proof). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ He showed the infinite series 1 / 2 + 1 / 3 + 1 / 5 + 1 / 7 + 1 / 11 + ... WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Euclid also showed how to construct a perfect number from a Mersenne prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ A = The set of prime number twins of the above form. Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
^ Primes of the form 2 p − 1, where p is a prime number, are known as Mersenne primes, while primes of the form are known as Fermat primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Prime numbers in nature . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x /log( x ), where log( x ) is the natural logarithm of x . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime, such as the work of Stern and Zeisel. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Ideas of Riemann in his 1859 paper on the zetafunction sketched a program which would lead to a proof of the prime number theorem. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ P. Hartmann, Prime number proofs . id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
This outline was completed by
Hadamard and
de la Vallée Poussin, who independently proved the
prime number theorem in 1896.
.^ Proving a number is prime is not done (for large numbers) by trial division. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Because these are the numbers easiest to prove prime! How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime, such as the work of Stern and Zeisel. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ [The LucasLehmer test is introduced. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ In 1891 Lucas turned Fermat's Little Theorem into a practical primality test. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ LucasLehmer test . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Edson Smith , the systems administrator at UCLA who found the largest Mersenne prime, explained that primes and even Mersenne primes are easy to find in the lower numbers, like 3 and 5, but become much more difficult to find when the numbers become long and intricate. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ I have worked on this before and my findings are that there almost seems to be a correlation between the DNA double helix and the Prime numbers algorithm. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ As a hint, there are faster but more complicated algorithms, like the Sieve of Atkin and the various Wheel Sieves. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
.^ The first 30 prime numbers are: . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Prime number algorithm in C . Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ Publickey cryptography . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Since 1951 all the largest known primes have been found by computers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ A grid of 75 computers at UCLA has found the largest prime number known to man . Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ This is also the seventh largest known prime of any form. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The search for ever larger primes has generated interest outside mathematical circles. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the selfinterest of studying the topic. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ So when conducting our search for these two primes, we only need to search circles BELOW the circle that 1225 is on, thereby eliminating an enormous amount of unnecessary search effort. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
.^ We all have limited lifetimes, and waiting a year for the computer to find primes to 220 trillion isn't rewarding.
^ So, even if you continued adding an infinite number of circles, you will only find prime numbers appearing somewhere along extensions of these 8 rays ! Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ I envision the main computer as running the prime finding program, but also replying to requests from satellites.
The number of prime numbers
.^ There are many open questions about prime numbers. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ It is conjectured there are infinitely many primes of the form n 2 + 1. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Functionalism Euclid and The Scientific Method of Today Some historians credit an ancient Greek mathematician known as Euclid with implementing logical processes for research and theoretical development, which would later evolve into the scientific method that is universally taught and used today.
^ The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:
.^ Consider any finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ In other words, when considering the set of integers as a ring, − 7 is a prime element. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Because all nonprime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ Multiply all of them together and add one (see Euclid number). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ P1P2 represents the product obtained by multiplying the two prime numbers together. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ When you multiply any two numbers together, it's usually fairly trivial to take that final answer and work backwards to figure out the original two numbers that were used. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
.^ Either way, there is at least one more prime that was not in the finite set we started with. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Because all nonprime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ So let us (me) list the prime numbers (a number that is only dividable by 1 and the number itself). Another game: Prime numbers and twins.  WebProWorld 28 January 2010 1:41 UTC www.webproworld.com [Source type: General]
.^ So there are more primes than any given finite number. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Of course, there are even more exotic ways to store primes.
^ Either way, there is at least one more prime that was not in the finite set we started with. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ This argument applies no matter what finite set we began with. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Note that there is no finite set of bases that will work in Miller's test. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following: . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Counting the number of prime numbers below a given number . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
(Euclid,
Elements: Book IX, Proposition 20)
.^ This previous argument explains why the product P of finitely many primes plus 1 must be divisible by some prime not among those finitely many primes (possibly itself). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Because all nonprime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Adding 1 to this product will always produce an even number, which will be divisible by 2 (and therefore not be prime). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ "I am simply sharing that the skips found in the lines are always a product of a prime times another prime" Yes, that is the Fundamental Theorem of Arithmetic which Euclid proved sometime before Jesus was born. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ Actually, Euclid offered a proof of the infinity of primes around 2000 years ago ... Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
.^ Consider any finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Adding the reciprocals of all primes together results in a divergent infinite series (proof). WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following: . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ In summary therefore, and as far as I can see, the initial five primes may actually be 1, 5, 7, 11, 13 and NOT the conventional 1, 2, 3, 5 and 7. Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
^ For any unique factorization domain, such as the ring Z of integers, the set of prime elements equals the set of irreducible elements, which for Z is {..., −11, −7, −5, −3, −2, 2, 3, 5, 7, 11, ... WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Because all nonprime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
That conclusion relies on a hypothesis later proved false, and so cannot be considered proved. The smallest counterexample with composite
P + 1 is
 (2 × 3 × 5 × 7 × 11 × 13) + 1 = 30,031 = 59 × 509 (both primes).
.^ Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following: . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Main article: formula for primes There is no known formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers". WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Actually, Euclid offered a proof of the infinity of primes around 2000 years ago ... Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers), page 1 28 January 2010 1:41 UTC www.abovetopsecret.com [Source type: FILTERED WITH BAYES]
Adding the reciprocals of all primes together results in a divergent
infinite series:
The
proof of that statement is due to
Euler. More precisely, if
S(
x) denotes the sum of the reciprocals of all prime numbers
p with
p ≤
x, then
 S(x) = ln ln x + O(1) for x → ∞.
.^ Of course, as far as proofs go, this theorem is only useful for proving that a given number is composite. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^{[12]} .^ Kummer's is particularly elegant and Harry Furstenberg provides one using general topology. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^{[14]}
.^ There are infinitely many prime numbers . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ It is not known whether there are infinitely many primorial or factorial primes. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ It is conjectured there are infinitely many primes of the form n 2 + 1. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
.^ This shows that there exist infinitely many prime numbers." id:A000040  OEIS Search Results 28 January 2010 1:41 UTC www.research.att.com [Source type: Academic]
^ Therefore, there exist gaps between primes which are arbitrarily large, i.e. WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ The Fundamental Theorem of Arithmetic states that for every number, there is exactly one way to factor that number into primes  and vice versa: every selection of primes multiplies into a different number. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^{[15]}
Verifying primality
Main article:
Primality test
.^ Counting the number of prime numbers below a given number . WikiSlice 28 January 2010 1:41 UTC dev.laptop.org [Source type: Reference]
^ Already some people are using keys that, in order to factor with the Number Field Sieve, would require more energy than exists in the known universe. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Few are the mathematicians who study creatures like the prime numbers with the hope or even desire for their discoveries to be useful outside of their own domain. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ There are several ways around this problem.
.^ Your algorithm could be used as an alternative to Eratosthenes' sieve, if > > the purpose was to find out *all* the prime numbers smaller than N, but > > it would be overkill to test a single number for primeness. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ We all have limited lifetimes, and waiting a year for the computer to find primes to 220 trillion isn't rewarding.
^ Your algorithm could be used as an alternative to Eratosthenes' sieve, if the purpose was to find out *all* the prime numbers smaller than N, but it would be overkill to test a single number for primeness. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
.^ For very small primes we can use the Sieve of Eratosthenes or trial division . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Your algorithm could be used as an alternative to Eratosthenes' sieve, if > > the purpose was to find out *all* the prime numbers smaller than N, but > > it would be overkill to test a single number for primeness. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ Your algorithm could be used as an alternative to Eratosthenes' sieve, if the purpose was to find out *all* the prime numbers smaller than N, but it would be overkill to test a single number for primeness. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
The modern
sieve of Atkin is more complicated, but faster when properly optimized.
.^ Few are the mathematicians who study creatures like the prime numbers with the hope or even desire for their discoveries to be useful outside of their own domain. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Why was the process so significantly slower using prime factors up to a billion?
^ Let's try finding prime numbers to a billion using a list of primes from 2 to 5 million: .
^{[16]}
.^ Prime number algorithm in C . Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ There are several ways to determine whether or not a given number is a prime number. Prime Numbers  Dev Shed 28 January 2010 1:41 UTC forums.devshed.com [Source type: General]
^ Two more pieces of advice which could help if you are running this more than once at a time: You only actually need to check against primes, if you know the primes less than the number already. Prime Numbers  Dev Shed 28 January 2010 1:41 UTC forums.devshed.com [Source type: General]
.^ And all numbers in this pattern cannot divided by 2 or 3. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ To test n for primality (to see if it is prime) just divide by all of the primes less than the square root of n . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ My logic is that I start a loop of all the numbers between the start and end, and inside that loop I take every number below my number from the first loop, and divide them. Prime Number Program 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
.^ When it comes to memory usage, the quickest and most obvious improvement comes with the realization that each candidate number either is or is not prime, and that's all you need to know.
^ We learned that the frequency of prime numbers decreases as the numbers go up, but they decrease at a decreasing rate, leveling off somewhere around an average of 1 out of 25.
^ From there, if you really enjoy prime numbers, you can venture out on the net to find truly optimized algorithms.
Otherwise, it is a prime.
.^ SPRP, 121 = 11.11 is a 3SPRP, 781 = 11.71 is a 5SPRP and, 25 = 5.5 is a 7SPRP. A test based on these results is quite fast, especially when combined with trial division by the first few primes. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ To find individual small primes trial division works well. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ If you pick a value for N that is divisible by 2 or 3 (the prime factors of 12), then you will find that you will only hit certain numbers before you return to midnight, and the sequence will then repeat. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ The prime factors of a given number (n) cannot be greater than ceiling (√n). Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ Consecutive numbers, increasing quantity of prime factors . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
.^ A number is therefore defined by its prime factorization. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ The largest prime number yet discovered has just been revealed to the world. Multi Prime  A multithreaded prime number benchmark  EXTREME Overclocking Forums 28 January 2010 1:41 UTC forums.extremeoverclocking.com [Source type: FILTERED WITH BAYES]
^ Prime numbers less than 2^18 . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
.^ There are more interesting directions to go. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ Etc… Therefore, there is a speed increase because one only needs to mark certain items. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
.^ MR 58:470a Monier80 L. Monier , "Evaluation and comparsion of two efficient probablistic primality testing algorithms," Theoretical Computer Science , 12 :1 (1980) 97108. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ To make a quick primality test from these results, start by dividing by the first few primes (say those below 257); then perform strong primality tests base 2, 3, ... How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ These probable primality tests can be combined to create a very quick algorithm for proving primality for integers less than 340,000,000,000,000. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ On the other hand, if p > 1 is composite, then it has a prime divisor q . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Prime number algorithm in C . Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ So I decided to try to convert the Sieve of Eratosthenes to TSQL. This algorithm is known to be both simple and fast for getting a list of prime numbers. SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
.^ Prime numbers less than 2^18 . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Prime number algorithm in C . Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ It is so obvious and the area of prime numbers is one of the most studied. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
.^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Here's the chunk of my code that checks for prime numbers. Prime Number Program 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ In addition, the order in which you visit the numbers is entirely dependent on what value you pick for N. In a similar vein, it is important that both P and Q be relatively prime to phi(R). Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ If it has none, try a Fermat test to see if it is a probable prime. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Strong probableprimality and a practical test . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ In 1891 Lucas turned Fermat's Little Theorem into a practical primality test. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
Thus, if
 a^{p − 1} (mod p)
is unequal to 1,
p is definitely composite.
.^ Here is the bad news: repeated PRP tests of a Carmichael number will fail to show that it is composite until we run across one of its factors. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Definition: The composite integer n is a Carmichael number if a n 1 =1 (mod n ) for every integer a relatively prime to n . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probableprimes. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ HI, is this is this solution to test if a number is a prime number or not: /* * Is n a prime number? Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ Chapter 2: The quick tests for small numbers and probable primes . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Chapter Two: The quick tests for small numbers and probable primes . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
However, the most popular probabilistic tests do not suffer from this drawback. The following table compares some primality tests.
.^ MR2123939 Abstract: We present a deterministic polynomialtime algorithm that determines whether an input number n is prime or composite. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ The definition of the function may be to test a single number for primeness, but it may be called many times for many numbers. Prime number algorithm in C  C / C++ answers 28 January 2010 1:41 UTC bytes.com [Source type: FILTERED WITH BAYES]
^ Agrawal, Kayal and Saxena managed to reformulate this into the following algorithm which they proved would run in at most O((log n ) 12 f (log log n )) time where f is a polynomial. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
Special types of primes
Construction of a regular pentagon. 5 is a Fermat prime.
.^ What he is doing is what we used to cal "digital factoring" (before that came to mean something else), wherein he is using the decimal digits of the number to calculate "digital roots" for various primes which shortcut methods for determining their remainder modulo that prime. SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
^ (There are more examples on the glossary page " probable prime ". How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Mersenne prime numbers are a class of primes named after Marin Mersenne , a 17th century French monk who studied the rare numbers 300 years ago. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ So I decided to try to convert the Sieve of Eratosthenes to TSQL. This algorithm is known to be both simple and fast for getting a list of prime numbers. SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
^ The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Finding Very Small Primes Fermat, ProbablePrimality and Pseudoprimes Strong ProbablePrimality and a Practical Test This is one of four chapters on finding primes and proving primality. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ The Neoclassical Tests, especially APR and APRCL Using Elliptic Curves, especially the ECPP Test A Polynomial Time Algorithm Conclusion and Suggestions This is one of four chapters on finding primes and proving primality. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
 n = 2^{i} · m
where
.^ (By the way, this is one of the reasons that 1 is not considered to be a prime number: if it were, then each number would have an infinite number of prime factorizations, all differing by how many 1s were included. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ R has to be the product of two prime numbers, don't forget. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ The nonprimepattern is that all the nonprimes are products of two numbers. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
.^ So I decided to try to convert the Sieve of Eratosthenes to TSQL. This algorithm is known to be both simple and fast for getting a list of prime numbers. SELECT Hints, Tips, Tricks FROM Hugo Kornelis WHERE RDBMS = 'SQL Server' : The prime number challenge – great waste of time! 28 January 2010 1:41 UTC sqlblog.com [Source type: FILTERED WITH BAYES]
^ One obvious improvement would be to test each candidate only with numbers that have been already proven prime.
^ A prime number is a whole number that can be divided only by one and itself. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
A prime
p is called
primorial or
primefactorial if it has the form
 p = n# ± 1
for some number
.^ Danny, if this stands for larger primes, or *all* primes, there is no way in the world you would not be known for eternity. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ However, your claim that all the skips in these columns are the product of two primes is off. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
^ The nonprimepattern is that all the nonprimes are products of two numbers. Prime Numbers  There is a pattern!  Danny Cooper  Blogs 28 January 2010 1:41 UTC www.aspose.com [Source type: FILTERED WITH BAYES]
A prime is called
factorial if it is of the form
n! ± 1.
.^ Primes associated to Primorials and Factorials . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ (By the way, this is one of the reasons that 1 is not considered to be a prime number: if it were, then each number would have an infinite number of prime factorizations, all differing by how many 1s were included. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
Location of the largest known prime
.^ [The LucasLehmer test is introduced. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ There are a handful of numbers which pass this test for every base, but which are not prime. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ These tests have been used for over 99.99% of the largest known primes. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ At this site we keep a list of the 5000 largest known primes, so if you do find new record primes, why not let us know ? How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Have you ever looked at the list of largest known primes ? How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
Prime 
Number of decimal digits 
Type 
Date 
Found by 
2^{43,112,609} − 1 
12,978,189 
Mersenne prime 
August 23, 2008 
Great Internet Mersenne Prime Search 
19,249 × 2^{13,018,586} + 1 
3,918,990 
not a Mersenne prime (Proth number) 
March 26, 2007 
Seventeen or Bust 
392113# + 1 
169,966 
primorial prime 
2001 
Heuer^{[17]} 
34790! − 1 
142,891 
factorial prime 
2002 
Marchal, Carmody and Kuosa ^{[18]} 
65516468355 × 2^{333333} ± 1 
100,355 
twin primes 
2009 
Twin prime search^{[19]} 
.^ If no factors are found, it's prime.
^ A grid of 75 computers at UCLA has found the largest prime number known to man . Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ Mersenne prime numbers are a class of primes named after Marin Mersenne , a 17th century French monk who studied the rare numbers 300 years ago. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
.^ As a prize, the Electronic Frontier Foundation is handing out $100,000, with half going to the winner and half going to charity. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ By the time you hit that point, 4.67 seconds to decode 6 million text primes is the least of your concerns.
^ To test this hypothesis, let's see how fast it produces 100,000,000 instead of a million, once again using pages of 100,000.
.^ As a prize, the Electronic Frontier Foundation is handing out $100,000, with half going to the winner and half going to charity. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ The discovery is part of the Great Internet Mersenne Prime Search (GIMPS), a 12yearold project that uses the computers of volunteers to find larger and larger prime numbers. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ If you are aiming for the money, then either join GIMPS (as Mersenne's have held te record for quite awhile now) or look for a very large generalized Fermat. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^{[20]} .^ First I ran the nonpaging program to produce all primes in the range 2 to 100,000,000 (a hundred million).
^ To test this hypothesis, let's see how fast it produces 100,000,000 instead of a million, once again using pages of 100,000.
^ GIMPS founder George Woltman said in a press release that the organization next will offer up a $150,000 award for the first person or group to find the first 100milliondigit prime number. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^{[21]}
Generating prime numbers
.^ Prime squares composed by no more than two digits . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Already some people are using keys that, in order to factor with the Number Field Sieve, would require more energy than exists in the known universe. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ A prime number is a whole number that can be divided only by one and itself. Grid power: Sysadmin discovers 13milliondigit prime number 28 January 2010 1:41 UTC www.computerworld.com [Source type: General]
^ In fact, for all ranges below one trillion, we'd use only primes in the 0 to one million range.
^ Euler's Totient Function is denoted by the Greek letter phi, and is defined as follows: phi(N) = how many numbers between 1 and N  1 which are relatively prime to N. Thus: . Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ In fact, for all ranges below one trillion, we'd use only primes in the 0 to one million range.
^ I started off using the formulas Joe suggested relating to primes but found I kept coming up with the same or simlar answers to those already posted. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ Next I ran the nonpaging program to produce all primes 2 to 1,000,000 (one million), saving the results to 0.pri , which is the prime factor input file used by the paging program.
.^ If you pick a value for N that is divisible by 2 or 3 (the prime factors of 12), then you will find that you will only hit certain numbers before you return to midnight, and the sequence will then repeat. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Until then, at least we have learned that there is a polynomialtime algorithm for all integers that both is deterministic and relies on no unproved conjectures! How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ This method is so fast that there is no reason to store a large list of primes on a computeran efficient implementation can find them faster than a computer can read from a disk. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Again all integers n > 1 which fail this test are composite; integers that pass it might be prime. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ Until then, at least we have learned that there is a polynomialtime algorithm for all integers that both is deterministic and relies on no unproved conjectures! How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ In 2002 a long standing question was answered: can integers be prove prime in "polynomial time" (that is, with time bounded by a polynomial evaluated at the number of digits). How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
.^ Euler's Totient Function is denoted by the Greek letter phi, and is defined as follows: phi(N) = how many numbers between 1 and N  1 which are relatively prime to N. Thus: . Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ On the other hand, 17, which is prime, results in 1 every time: . Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ An illustration: At the time of my writing, one of the largest general numbers that has been independently factored was the number used as the modulus for the RSA140 challenge. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ The paging program in this article uses the output of the page 0 producing program as its input  in other words, as its prime factors.
^ It is guaranteed that there is no other way to break 1176 into prime factors. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
Distribution
Given the fact that there is an infinity of primes, it is natural to seek for patterns or irregularities in the distribution of primes.
.^ MR 2000e:11160 HL23 G. H. Hardy and J. E. Littlewood , "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ No memory problems, no need to preestimate the number of primes.
^ A new class of prime numbers that involve the popular Google search engine is explained.
.^ For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens? How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ From there, if you really enjoy prime numbers, you can venture out on the net to find truly optimized algorithms.
^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
Leonhard Euler commented
.^ Already some people are using keys that, in order to factor with the Number Field Sieve, would require more energy than exists in the known universe. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ If it's not too late, here it is: INSERT INTO Primes (p) SELECT s1.seq FROM Sequence s1 WHERE NOT EXISTS (SELECT 1 FROM Sequence s2 WHERE s2.seq BETWEEN 2 AND SQRT (s1.seq) AND s1.seq % s2.seq = 0) . Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ Few are the mathematicians who study creatures like the prime numbers with the hope or even desire for their discoveries to be useful outside of their own domain. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^{[22]}
.^ There are two useful facts from Number Theory: . Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ Two dice to produce prime numbers . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ What about the 200 trillion you could run in a year  how many prime factors would you need?
.^ Prime numbers less than 2^18 . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ First primes embedded in the smallest number . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
.^ Of course, there are even more exotic ways to store primes.
^ Called Carmichael numbers , they are far more rare than the prime numbers  but, like the primes numbers, there are still an infinite number of them. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^ Few are the mathematicians who study creatures like the prime numbers with the hope or even desire for their discoveries to be useful outside of their own domain. Prime Number HideandSeek: How the RSA Cipher Works 28 January 2010 1:41 UTC www.muppetlabs.com [Source type: FILTERED WITH BAYES]
^{[23]}
Euler noted that the function
 n^{2} + n + 41
gives prime numbers for
n < 40 (but not necessarily so for bigger
n), a remarkable fact leading into deep
algebraic number theory, more specifically
Heegner numbers.
.^ Integers The Way of the Locust: Reflections on the Nature of Intelligence and Information A look at the locust and how it knows primes numbers!
Surprisingly, prime numbers cluster on certain diagonals and not others.
The number of prime numbers below a given number
.^ We learned that the frequency of prime numbers decreases as the numbers go up, but they decrease at a decreasing rate, leveling off somewhere around an average of 1 out of 25.
^ Function test prime tod etermine if number is prime or not. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ This makes a lot of sense, because as candidate numbers go up, marking more numbers as composite, and therefore lessening the frequency of primes.
.^ I think it would be beneficial to have another column in the sequence table, that houses Prime candidates, so you can not process numbers that you know will not be Prime, such as even numbers greater than 2. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ At the extreme, if the list of “MOD (seq, )” expressions goes to a value equal or higher than the upper limit we are looking at, we get the answer immediately. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ These probable primality tests can be combined to create a very quick algorithm for proving primality for integers less than 340,000,000,000,000. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
Values as large as π(10
^{20}) can be calculated quickly and accurately with modern computers.
.^ Prime factors k · 2 n + 1 of larger Fermat numbers F m .
^ If a modulo value exists between the current number and the lesser number, the number is not prime. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ No memory problems, no need to preestimate the number of primes.
.^ We can test 100,000,000 numbers for primeness in less than a minute.
^ My observation in the 100,000,000 neighborhood is that about 1/20 of the numbers are prime, and that as the range gets higher, the frequency of primes decreases slightly.
^ In other words, to find all primes under 1,000,000, we create an array of char 1,000,001 long, filled with ones.
.^ The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list . How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ I think it would be beneficial to have another column in the sequence table, that houses Prime candidates, so you can not process numbers that you know will not be Prime, such as even numbers greater than 2. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ Function test prime tod etermine if number is prime or not. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
.^ Space Astronomy The Goldbach Conjecture We are unsure about whether prime numbers greater than two can be expressed as the sum of two primes.
^ Hi, i'm a c beginner and have to write a programme to count the number of prime numbers less than 100,1000,10000,100000,1000000 respectively. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ In fact, I feel that everything should be type int, since "prime number" only makes since with integers. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
Gaps between primes
A sequence of consecutive integers none of which is prime constitutes a
prime gap.
.^ Prime numbers less than 2^18 . The Prime Puzzles and Problems Connection 16 September 2009 1:47 UTC www.primepuzzles.net [Source type: Reference]
^ Of course u and v must each be larger than the factoring bound B. With the above notation we can now state our final classical theorems. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ In previous sections we have pointed out if the factored portion of n 1 or of n +1 is larger than the cube root of n , then we can prove n is prime. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
read
factorial)
 n! + 2, n! + 3, …, n! + n
is a sequence of n − 1 consecutive composite integers, since
 n! + m = m · (n!/m + 1) = m · [(1 · 2 · … · (m − 1) · (m + 1) … n) + 1]
is composite for any 2 ≤ m ≤ n. On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient
 (p_{i + 1} − p_{i}) / p_{i},
where
p_{i} denotes the
ith prime number (i.e.,
p_{1} = 2,
p_{2} = 3, etc.),
approaches zero as
i approaches infinity.
Open questions
The Riemann hypothesis
.^ Now the first number left is 5, the second odd primecross out all of its multiples. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens? How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
^ It took about 24 seconds to write out the primes in the first 100,000,000 numbers, and it took less than 5 seconds to read, convert, convert and write them back.
For example, the fact (see above) that there are infinitely many primes can be read off from the divergence of the
harmonic series:
Riemann's hypothesis is concerned with the zeroes of the ζfunction (i.e.,
s such that ζ(
s) = 0).
.^ For instance, let's say we're testing prime candidates in pages of one million, and we're now testing numbers between one million and two million.
From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise.
.^ We can test 100,000,000 numbers for primeness in less than a minute.
^ Or any number multiplied by a number greater than the square root of the limit... Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ The prime factors of a given number (n) cannot be greater than ceiling (√n). Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
This hypothesis is generally believed to be correct.
.^ This method is so fast that there is no reason to store a large list of primes on a computeran efficient implementation can find them faster than a computer can read from a disk. How to find primes and prove primality (merged version) 16 September 2009 1:47 UTC primes.utm.edu [Source type: Academic]
Other conjectures
.^ What about the 200 trillion you could run in a year  how many prime factors would you need?
^ When it comes to memory usage, the quickest and most obvious improvement comes with the realization that each candidate number either is or is not prime, and that's all you need to know.
^ This makes a lot of sense, because as candidate numbers go up, marking more numbers as composite, and therefore lessening the frequency of primes.
.^ If a modulo value exists between the current number and the lesser number, the number is not prime. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
It is conjectured that there are infinitely many
Fibonacci primes^{[24]} and infinitely many
Mersenne primes, but not
Fermat primes.
^{[25]} .^ It is known from Dirichlet that a + nd, where n ≥ 0 series contains infinite number of primes. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ From there, if you really enjoy prime numbers, you can venture out on the net to find truly optimized algorithms.
.^ Christian Goldbach conjectured on this aspect of prime number theory.
^ Space Astronomy The Goldbach Conjecture We are unsure about whether prime numbers greater than two can be expressed as the sum of two primes.
It is conjectured that there are infinitely many
twin primes, pairs of primes with difference 2 (
twin prime conjecture).
Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer
n, there are infinitely many pairs of consecutive primes which differ by 2
n.
.^ We could make 2 a special case and just print it, not include it in the list of primes, start our candidates with 3, and increment by 2 every time.
^ These 2 primes comes from the n=0 cases of the 6n+2 and 6n+3 series. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^{.April 2009" style="whitespace:nowrap;">[citation needed]} Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2.
Legendre's conjecture states that there is a prime number between
n^{2} and (
n + 1)
^{2} for every positive integer
n.
^ My logic is that I start a loop of all the numbers between the start and end, and inside that loop I take every number below my number from the first loop, and divide them. Prime Number Program 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ For instance, let's say we're testing prime candidates in pages of one million, and we're now testing numbers between one million and two million.
^ Hi, i'm a c beginner and have to write a programme to count the number of prime numbers less than 100,1000,10000,100000,1000000 respectively. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
It is implied by the stronger
Cramér's conjecture.
.^ Christian Goldbach conjectured on this aspect of prime number theory.
^ Space Astronomy The Goldbach Conjecture We are unsure about whether prime numbers greater than two can be expressed as the sum of two primes.
^ Hi, i'm a c beginner and have to write a programme to count the number of prime numbers less than 100,1000,10000,100000,1000000 respectively. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
Applications
.^ Christian Goldbach conjectured on this aspect of prime number theory.
^ Prime Numbers Roots of Geometry: Euclid Gives an explanation of Euclid's life and his contribution to modern mathematics.
^ Prime Numbers Late August Offers Prime Planetary Views, Photo Opportunities Late August is a prime time for viewing and photographing the planets of the solar system.
In particular, number theorists such as
British mathematician
G. H. Hardy prided themselves on doing work that had absolutely no military significance.
^{[27]} .^ Let's try finding prime numbers to a billion using a list of primes from 2 to 5 million: .
^ Indeed, the easiest way to do the first 200 billion is to start with the standard bit array paging algorithm using a factor array of 15 million primes.
^ The prime number algorithms you find in this document won't break any records.
.^ Let's try finding prime numbers to a billion using a list of primes from 2 to 5 million: .
^ I think it would be beneficial to have another column in the sequence table, that houses Prime candidates, so you can not process numbers that you know will not be Prime, such as even numbers greater than 2. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
^ For the first attempt, let’s load the Primes table with candidate numbers using math fact #2 from above. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
.^ When it comes to memory usage, the quickest and most obvious improvement comes with the realization that each candidate number either is or is not prime, and that's all you need to know.
^ In other words, to find all primes under 1,000,000, we create an array of char 1,000,001 long, filled with ones.
^ The factors could be stored either in a different bitarray, or in an array of prime numbers.
This helped generate the
full cycle of possible rotor positions before repeating any position.
.^ Here's the chunk of my code that checks for prime numbers. Prime Number Program 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ That means if the machines divide the work evenly (big if), the ten machines can store 2 trillion proven prime numbers before human intervention becomes necessary.
^ For the first attempt, let’s load the Primes table with candidate numbers using math fact #2 from above. Celko's Summer SQL Stumpers: Prime Numbers 28 January 2010 1:41 UTC www.simpletalk.com [Source type: FILTERED WITH BAYES]
Arithmetic modulo a prime p
Modular arithmetic is a modification of usual arithmetic, by doing all calculations "modulo" a fixed number
n. All calculations of modular arithmetic take place in the
finite set
 {0, 1, 2, ..., n − 1}.
Calculating modulo
n means that sums, differences and products are calculated as usual, but then only the
remainder after division by
n is considered. For example, let
n = 7. Then, in modular arithmetic modulo 7, the sum 3 + 5 is 1 instead of 8, since 8 divided by 7 has remainder 1. Similarly, 6 + 1 = 0 modulo 7, 2 − 5 = 4 modulo 7 (since −3 + 7 = 4) and 3 · 4 = 5 modulo 7 (12 has remainder 5). Standard properties of
addition and
multiplication familiar from the number system of the
integers or
rational numbers remain valid, for example
 (a + b) · c = a · c + b · c (law of distributivity).
In general it is, however, not possible to divide in this setting. For example, for n = 6, the equation
 3 · x = 2 (modulo 6),
a solution
.^ How would the calculations go with 15 million prime factors? See the following: .
^ Solve Beautiful Numbers A simple rendition of what one can say about math.
^ In fact, for all ranges below one trillion, we'd use only primes in the 0 to one million range.
For
n = 7, the equation
 3 · x = 2 (modulo 7)
The set
{0, 1, 2, ..., n − 1}, with addition and multiplication is denoted
Z/
nZ for all
n. In the parlance of
abstract algebra, it is a
ring, for any
n, but a
finite field if and only if
n is prime. A number of theorems can be derived from inspecting
Z/
pZ in an abstract way. For example
Fermat's little theorem, stating that
a^{p} −
a is divisible by
p for any
integer a, may be proved using these notions.
.^ It took about 24 seconds to write out the primes in the first 100,000,000 numbers, and it took less than 5 seconds to read, convert, convert and write them back.
^ Hi, i'm a c beginner and have to write a programme to count the number of prime numbers less than 100,1000,10000,100000,1000000 respectively. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^ That's because a number bigger than the square of the highest test factor could have factors above the topmost prime factor.
.^ I believe that because prime frequency seems to approach 1 in 25 rather than 1 in 65535, consecutive primes varying by more than 65535 would be exceedingly rare.
^ That's because a number bigger than the square of the highest test factor could have factors above the topmost prime factor.
Wilson's theorem says that an integer
p > 1 is prime if and only if the
factorial (
p − 1)! + 1 is divisible by
p. Moreover, an integer
n > 4 is composite if and only if (
n − 1)! is divisible by
n.
Other mathematical occurrences of primes
.^ Let's try finding prime numbers to a billion using a list of primes from 2 to 5 million: .
^ This makes a lot of sense, because as candidate numbers go up, marking more numbers as composite, and therefore lessening the frequency of primes.
^ In fact, I feel that everything should be type int, since "prime number" only makes since with integers. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
An example from the theory of
finite groups are the
Sylow theorems: if
G is a finite group and
p^{n} is the
highest power of the prime p which divides the
order of
G, then
G has a subgroup of order
p^{n}. Also, any group of prime order is cyclic (
Lagrange's theorem).
Publickey cryptography
Several publickey cryptography algorithms, such as
RSA and the
DiffieHellman key exchange, are based on large prime numbers (for example with 512
bits). They rely on the fact that it is thought to be much easier (i.e., more efficient) to perform the multiplication of two (large) numbers
x and
y than to calculate
x and
y (assumed
coprime) if only the product
xy is known.
Prime numbers in nature
Inevitably, some of the numbers that occur in nature are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime.
One example of the use of prime numbers in nature is as an evolutionary strategy used by
cicadas of the genus
Magicicada.
^{[28]} These insects spend most of their lives as
grubs underground. They only pupate and then emerge from their burrows after 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most.
.^ In fact, I feel that everything should be type int, since "prime number" only makes since with integers. Prime Number Counting. Help!! 28 January 2010 1:41 UTC p2p.wrox.com [Source type: General]
^{[29]} If
Magicicadas appeared at a nonprime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200year period, average predator populations during hypothetical outbreaks of 14 and 15year cicadas would be up to 2% higher than during outbreaks of 13 and 17year cicadas.
^{[30]} Though small, this advantage appears to have been enough to drive natural selection in favour of a primenumbered lifecycle for these insects.
There is speculation that the zeros of the
zeta function are connected to the energy levels of complex quantum systems.
^{[31]}
Generalizations
The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the
prime field is the smallest subfield of a field
F containing both 0 and 1. It is either
Q or the
finite field with
p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in
knot theory, a
prime knot is a
knot which is indecomposable in the sense that it cannot be written as the
knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.
^{[32]} Prime models and
prime 3manifolds are other examples of this type.
Prime elements in rings
Prime numbers give rise to two more general concepts that apply to elements of any
ring R, an
algebraic structure where addition, subtraction and multiplication are defined:
prime elements and
irreducible elements. An element
p of
R is called prime if it is not a
unit (i.e., does not have a
multiplicative inverse) and the following property holds: given
x and
y in
R such that
p divides the product, then
p divides at least one factor. Irreducible elements are ones which cannot be written as a product of two ring elements that are not units. In general, this is a weaker condition, but for any
unique factorization domain, such as the ring
Z of integers, the set of prime elements equals the set of irreducible elements, which for
Z is :
 {…, −11, −7, −5, −3, −2, 2, 3, 5, 7, 11, …}.
A common example is the
Gaussian integers Z[
i], that is, the set of complex numbers of the form
a +
bi with
a and
b in
Z. This is an integral domain, its prime elements are known as
Gaussian primes. Not every prime (in
Z) is a Gaussian prime: in the bigger ring
Z[
i], 2 factors into the product of the two Gaussian primes (1 +
i) and (1 −
i). Rational primes (i.e. prime elements in
Z) of the form 4
k + 3 are Gaussian primes, whereas rational primes of the form 4
k + 1 are not. Gaussian primes can be used in proving
quadratic reciprocity, while
Eisenstein primes play a similar role for
cubic reciprocity.
Prime ideals
Main article:
Prime ideals
In
ring theory, the notion of number is generally replaced with that of
ideal.
Prime ideals, which generalize prime elements in the sense that the
principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in
commutative algebra,
algebraic number theory and
algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the
LaskerNoether theorem which expresses any ideal in a
Noetherian commutative ring as the intersection of
primary ideals, which are the appropriate generalizations of
prime powers.
^{[33]}
Primes in valuation theory
In algebraic number theory, yet another generalization is used. A starting point for
valuation theory is the
padic valuations, where
p is a prime number. It tells what highest power
p divides a given number
n. Using that, the
padic norm is set up, which, in contrast to the usual
absolute value, gets smaller when a number is
multiplied by
p. The
completion of
Q (the field of rational numbers) with respect to this norm leads to
Q_{p}, the field of
padic numbers, as opposed to
R, the reals, which are the completion with respect to the usual absolute value. To highlight the connection to primes, the absolute value is often called the
infinite prime. These are essentially all possible ways to complete
Q, by
Ostrowski's theorem.
In an arbitrary
field K, one considers
valuations on
K, certain functions from
K to the real numbers
R. Every such valuation yields a
topology on K, and two valuations are called equivalent if they yield the same topology. A
prime of K (sometimes called a
place of K) is an
equivalence class of valuations.
Arithmetic questions related to,
global fields such as
Q may, in certain cases, be transferred back and forth to the completed fields (known as
local fields), a concept known as
localglobal principle. This again underlines the importance of primes to number theory.
In the arts and literature
Prime numbers have influenced many artists and writers. The French
composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as
La Nativité du Seigneur (1935) and
Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in one of the études. According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".
^{[34]}
In his science fiction novel
Contact, later made into a
film of the same name, the
NASA scientist
Carl Sagan suggested that prime numbers could be used as a means of communicating with aliens, an idea that he had first developed informally with American astronomer
Frank Drake in 1975.
^{[35]}
Many films reflect a popular fascination with the mysteries of prime numbers and cryptography: films such as
Cube,
Sneakers,
The Mirror Has Two Faces and
A Beautiful Mind, the latter of which is based on the biography of the mathematician and Nobel laureate
John Forbes Nash by
Sylvia Nasar.
^{[36]} Prime numbers are used as a metaphor for loneliness and isolation in the
Paolo Giordano novel
The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.
^{[37]}
See also
Distributed computing projects that search for primes
Notes
 ^ (sequence A000040 in OEIS).
 ^ http://primes.utm.edu/notes/proofs/infinite/euclids.html
 ^ GIMPS Home; http://www.mersenne.org/
 ^ Riesel 1994, p. 36
 ^ Conway & Guy 1996, pp. 129–130
 ^ Derbyshire, John (2003). "The Prime Number Theorem". Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington, D.C.: Joseph Henry Press. p. 33. ISBN 9780309085496. OCLC 249210614.
 ^ Gowers 2002, p. 118 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes."
 ^ ""Why is the number one not prime?"". Retrieved 20071002.
 ^ ""Arguments for and against the primality of 1".
 ^ The Largest Known Prime by Year: A Brief History Prime Curios!: 17014…05727 (39digits)
 ^ Hardy 1908, pp. 122–123
 ^ Letter in Latin from Goldbach to Euler, July 1730.
 ^ Ribenboim 2004, p. 4
 ^ Furstenberg 1955
 ^ (Ben Green & Terence Tao 2008).
 ^ (Lehmer 1909).
 ^ The Top Twenty: Primorial
 ^ The Top Twenty: Factorial
 ^ The Top Twenty: Twin Prime Search
 ^ "Record 12MillionDigit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. October 14, 2009. http://www.eff.org/press/archives/2009/10/140. Retrieved 20100104.
 ^ "EFF Cooperative Computing Awards". Electronic Frontier Foundation. http://www.eff.org/awards/coop. Retrieved 20100104.
 ^ Havil 2003, p. 163
 ^ Havil 2003, p. 171
 ^ Caldwell, Chris, The Top Twenty: Lucas Number at The Prime Pages.
 ^ E.g., see Guy 1981, problem A3, pp. 7–8
 ^ Weisstein, Eric W., "Landau's Problems" from MathWorld.
 ^ Hardy 1940 "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."
 ^ Goles, E., Schulz, O. and M. Markus (2001). "Prime number selection of cycles in a predatorprey model", Complexity 6(4): 3338
 ^ Paulo R. A. Campos, Viviane M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. (2004). "Emergence of Prime Numbers as the Result of Evolutionary Strategy". Phys. Rev. Lett. 93: 098107. doi:10.1103/PhysRevLett.93.098107. http://link.aps.org/abstract/PRL/v93/e098107. Retrieved 20061126.
 ^ "Invasion of the Brood". The Economist. May 6, 2004. http://economist.com/PrinterFriendly.cfm?Story_ID=2647052. Retrieved 20061126.
 ^ Ivars Peterson (June 28, 1999). "The Return of Zeta". MAA Online. http://www.maa.org/mathland/mathtrek_6_28_99.html. Retrieved 20080314.
 ^ Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.B Heidelberger Akad. Wiss. Math.Nat. Kl. 1949 (1949), 57–104.
 ^ Eisenbud 1995, section 3.3.
 ^ Hill, ed. 1995
 ^ Carl Pomerance, Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007
 ^ The music of primes, Marcus du Sautoy's selection of films featuring prime numbers.
 ^ "Introducing Paolo Giordano". Books Quarterly. http://www.wbqonline.com/feature.do?featureid=342.
References
 Conway, John Horton; Guy, Richard K. (1996), The Book of Numbers, New York: Copernicus, ISBN 9780387979939
 Crandall, Richard; Pomerance, Carl (2005), Prime Numbers: A Computational Perspective (2nd ed.), Berlin, New York: SpringerVerlag, ISBN 9780387252827
 Derbyshire, John (2003), Prime obsession, Joseph Henry Press, Washington, DC, MR1968857, ISBN 9780309085496
 Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: SpringerVerlag, MR1322960, ISBN 9780387942681
 Furstenberg, Harry (1955), "On the infinitude of primes", The American Mathematical Monthly 62: 353, doi:10.2307/2307043, ISSN 00029890, http://www.jstor.org/stable/2307043
 Green, Ben; Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167: 481–547, arΧiv:math.NT/0404188
 Gowers, Timothy (2002), Mathematics: A Very Short Introduction, Oxford University Press, ISBN 9780192853615
 Guy, Richard K. (1981), Unsolved Problems in Number Theory, Berlin, New York: SpringerVerlag, ISBN 9780387905938
 Havil, Julian (2003), Gamma: Exploring Euler's Constant, Princeton University Press, ISBN 9780691099835
 Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press, ISBN 9780521092272
 Hardy, Godfrey Harold (1940), A Mathematician's Apology, Cambridge University Press, ISBN 9780521427067
 Lehmer, D. H. (1909), Factor table for the first ten millions containing the smallest factor of every number not divisible by 2, 3, 5, or 7 between the limits 0 and 10017000, Washington, D.C.: Carnegie Institution of Washington
 Narkiewicz, Wladyslaw (2000), The development of prime number theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Berlin, New York: SpringerVerlag, ISBN 9783540662891
 Ribenboim, Paulo (2004), The little book of bigger primes, Berlin, New York: SpringerVerlag, ISBN 9780387201696
 Riesel, Hans (1994), Prime numbers and computer methods for factorization, Basel, Switzerland: Birkhäuser, ISBN 9780817637439
 Sabbagh, Karl (2003), The Riemann hypothesis, Farrar, Straus and Giroux, New York, MR1979664, ISBN 9780374250072
 du Sautoy, Marcus (2003), The Music of Primes website The music of the primes, HarperCollins Publishers, MR2060134, ISBN 9780066210704, http://www.musicoftheprimes.com/ The Music of Primes website
Further references
 Hill, Peter Jensen, ed. (1995), The Messiaen companion, Portland, Or: Amadeus Press, ISBN 9780931340956
 Kelly, Katherine E., ed. (2001), The Cambridge companion to Tom Stoppard, Cambridge University Press, ISBN 9780521645928
 Stoppard, Tom (1993), Arcadia, London: Faber and Faber, ISBN 9780571169344
External links
Prime number generators and calculators