From Wikipedia, the free encyclopedia
The
logarithmic scale can compactly represent the relationship among variously sized numbers.
This list compares various sizes of positive numbers, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the so called short scale which is used in English speaking countries, as well as a name in the long scale which is used in a series of countries that do not have English as their national language.
Smaller than 10^{−30}
- Computing: The number 5 × 10^{−324} is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value.
- Computing: The number 1.4 × 10^{−45} is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.
10^{−30}
(0.000000000000000000000000000001; 1000^{−10}; short scale: one nonillionth, long scale: one quintillionth)
- Mathematics: The probability in a game of bridge of all four players getting a single-suit hand is approximately 4.47×10^{−28}^{[1]}
10^{−27}
(0.000000000000000000000000001; 1000^{−9}; short scale: one octillionth, long scale: one quadrilliardth)
10^{−24}
(0.000000000000000000000001; 1000^{−8}; short scale: one septillionth long scale: one quadrillionth)
ISO: yocto- (y)
10^{−21}
(0.000000000000000000001; 1000^{−7}; short scale: one sextillionth, long scale: one trilliardth)
ISO: zepto- (z)
10^{−18}
(0.000000000000000001; 1000^{−6}; short scale: one quintillionth, long scale: one trillionth)
ISO: atto- (a)
- Mathematics: The probability of rolling snake eyes 10 times in a row on a pair of fair dice is about 2.74×10^{−16}
10^{−15}
(0.000000000000001; 1000^{−5}; short scale: one quadrillionth, long scale: one billiardth)
ISO: femto- (f)
10^{−12}
(0.000000000001; 1000^{−4}; short scale: one trillionth, long scale: one billionth)
ISO: pico- (p)
- Mathematics: The probability in a game of bridge of one player getting a single-suit hand is approximately 2.52×10^{−11} (0.00000000252%)
10^{−9}
(0.000000001; 1000^{−3}; short scale: one billionth; long scale: one milliardth)
ISO: nano- (n)
- Mathematics — Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the US Powerball lottery, with a single ticket, under the rules as of August 2009, are 195,249,053 to 1 against, for a probability of 5.12×10^{−9} (0.000000512%).
- Mathematics — Lottery: The odds of winning the Jackpot (matching the 6 main numbers) in the UK National Lottery, with a single ticket, under the rules as of August 2009, are 13,983,815 to 1 against, for a probability of 7.15×10^{−8} (0.00000715%).
10^{−6}
(0.000001; 1000^{−2}; long and short scales: one millionth)
ISO: micro- (μ)
- Mathematics - Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5 × 10^{−6} (0.00015%).
- Mathematics — Poker: The odds of being dealt a straight flush (other than a royal flush) in poker are 72,192 to 1 against, for a probability of 1.4 × 10^{−5} (0.0014%).
- Mathematics — Poker: The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4 × 10^{−4} (0.024%).
10^{−3}
(0.001; 1000^{−1}; one thousandth)
ISO: milli- (m)
- Mathematics — Poker: The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10^{−3} (0.14%).
- Mathematics — Poker: The odds of being dealt a flush in poker are 507.8 to 1 against, for a probability of 1.9 × 10^{−3} (0.19%).
- Mathematics — Poker: The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10^{−3} (0.39%).
- Physics: α = 0.007297352570(5), the fine-structure constant.
10^{−2}
(0.01; one hundredth)
ISO: centi- (c)
- BioMed — HIV: About 1.2% of all 15–49 year-old humans were infected with HIV at the end of 2001^{[citation needed]}
- Mathematics — Lottery: The odds of winning any prize in the UK National Lottery, with a single ticket, under the rules as of 2003, are 54 to 1 against, for a probability of about 0.018 (1.8%)
- Mathematics — Poker: The odds of being dealt a three of a kind in poker are 46 to 1 against, for a probability of 0.021 (2.1%)
- Mathematics — Lottery: The odds of winning any prize in the US Powerball Multistate Lottery, with a single ticket, under the rules as of 2006, are 36.61 to 1 against, for a probability of 0.027 (2.7%)
- Mathematics — Poker: The odds of being dealt two pair in poker are 20 to 1 against, for a probability of 0.048 (4.8%).
10^{−1}
(0.1; one tenth)
ISO: deci- (d)
- Mathematics — Poker: The odds of being dealt only one pair in poker are about 5 to 2 against (2.37 to 1), for a probability of 0.42 (42%).
- Mathematics — Poker: The odds of being dealt no pair in poker are nearly 1 to 2, for a probability of about 0.5 (50%)
10^{0}
(1; one)
10^{1}
(10; ten)
ISO: deca- (da)
- Human scale: there are 10 fingers on a pair of human hands, and 10 toes on a pair of human feet.
- Mathematics: the number system used in everyday life, the decimal system, has 10 digits: 0,1,2,3,4,5,6,7,8,9.
- Mathematics: the hexadecimal system, a common number system used in computer programming, uses 16 digits where the last 6 are usually represented by letters: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
- Language: there are 26 letters in the Latin alphabet in the English language.
10^{2}
(100; hundred)
ISO: hecto- (h)
- Computing: There are 128 characters in the ASCII character set.
- Political Science: There were 192 member states of the United Nations as of 2006.
10^{3}
(1000; thousand)
ISO: kilo- (k)
- Language: 2,000–3,000 letters on a typical typed page of text.
- BioMed: the DNA of the simplest viruses has some 5,000 base pairs.
- Language: There are about 6,500 mutually unintelligible languages and dialects.^{[citation needed]}
10^{4}
(10000; ten thousand or a myriad)
- BioMed: Each neuron in the human brain is estimated to connect to 10,000 others
- Language: There are 20,000–40,000 distinct Chinese characters, depending on how one counts them
- BioMed: Each human being is estimated to have 30,000 to 40,000 genes
- Records: As of July 2004, the largest number of decimal places of π that have been recited from memory - > 42000
- Mathematics: 65,537 is the largest known Fermat prime
10^{5}
(100000; one hundred thousand or a lakh)
- BioMed — Strands of hair on a head: The average human head has about 100,000–150,000 strands of hair
- Mathematics: 110,000 - The approximate number of entries on The On-Line Encyclopedia of Integer Sequences as of August 2005 ^{[2]}
- Language: 267,000 words in James Joyce's Ulysses
- Language — English words: The New Oxford Dictionary of English contains about 350,000 definitions for English words
- Language: 564,000 words in War and Peace
10^{6}
(1000000; 1000^{2}; long and short scales: one million)
ISO: mega- (M)
- BioMed — Species: The World Resources Institute claims that approximately 1.4 million species have been named, out of an unknown number of total species (estimates range between 2 and 100 million species)
- Info: The FreeDB database has around 1 750 000 entries as of June 2005
- Mathematics — Playing cards: There are 2 598 960 different 5-card poker hands that can be dealt from a standard 52-card deck.
- Info — Web sites: as of 26 February, 2010, Wikipedia contains approximately 3,206,530 articles in the English language
- Geography/Computing — Geographic places: The NIMA GEOnet Names Server contains approximately 3.88 million named geographical features outside the United States, with 5.34 million names. The USGS Geographic Names Information System claims to have almost 2 million physical and cultural geographic features within the United States
- Genocide: Approximately 6,000,000 Jews were killed in the Holocaust.
- Mathematics: 12,988,816 is the number of domino tilings of an 8×8 checkerboard.
- Info — Web sites: as of July 2003, the Netcraft web survey estimates that there are 42 million distinct web sites
- Info — Books: The British Library claims that it holds over 150 million items. The Library of Congress claims that it holds approximately 119 million items. See The Gutenberg Galaxy
- Mathematics: 215,000,000 - The approximate number of mathematical constants collected on the Plouffe's Inverter as of August 2005 [1]
- Mathematics: 275,305,224 is the number of 5x5 normal magic squares, not counting rotations and reflections. This result was found in 1973 by Richard Schroeppel. It is the third 91768409-gonal number.
- Mathematics: 358,833,097 stellations of the rhombic triacontahedron
- Demographics: approx. 402,000,000 native speakers of English
- Astronomy — Cataloged stars: The Guide Star Catalog II has entries on 998,402,801 distinct astronomical objects
10^{9}
(1000000000; 1000^{3}; short scale: one billion; long scale: one thousand million, or one milliard)
ISO: giga- (G)
- Demographics — India: 1,096,000,000 - approximate population of India in 2007
- Demographics — China: 1,311,000,000 - approximate population of the People's Republic of China in 2007.
- Computing — Computational limit of a 32-bit CPU: 2 147 483 647 is equal to 2^{31}−1, and as such is the largest number which can fit into a signed (two's complement) 32-bit integer on a computer, thus marking the upper computational limit of a 32-bit CPU such as Intel's Pentium-class computer chips.
- BioMed — base pairs in the genome: approximately 3 × 10^{9} base pairs in the human genome
- Computing - IPv4: 4,294,967,296 (2^{32}) possible unique IP addresses.
- Computing: 4,294,967,296 - the number of bytes in 4 gibibytes; in computation, the 32-bit computers can directly access 2^{32} pieces of address space, this leads directly to the 4 gigabyte limit on main memory.
- Mathematics: 4,294,967,297 is a Fermat number and semiprime. It is the smallest number of the form which is not a prime number.
- Demographics — world population: 6,587,890,000 - Estimated total mid-year population for the world in 2007 (April 10).
- Computing — web pages: approximately 8 × 10^{9} web pages indexed by Google as of 2004
- BioMed — bacteria in the human body: there are roughly 10^{10} bacteria in the human oral cavity ^{[3]}
- Astronomy — Observable galaxies: as of 2003 there are between 1 × 10^{10} and 8 × 10^{10} galaxies in the observable Universe
- Physical cosmology — Age of the universe: Current theory and observations suggest that approximately 1.4 × 10^{10} years have passed since the Big Bang.
- BioMed — Neurons in the brain: approximately 10^{11} neurons in the human brain
- Astronomy — stars in our galaxy: approximately 4 × 10^{11} stars in the Milky Way galaxy
10^{12}
(1000000000000; 1000^{4}; short scale: one trillion; long scale: one billion)
ISO: tera- (T)
- BioMed — Bacteria on the human body: the surface of the human body houses roughly 10^{12} bacteria.^{[4]}
- Mathematics — Known digits of π: as of 2002, the number of known digits of π was 1,241,100,000,000 (1.2411 × 10^{12}).
- Marine biology: 3,500,000,000,000 (3.5 × 10^{12}) - estimated population of fish in the ocean.
- Mathematics: 7,625,597,484,987 - a number that often appears when dealing with powers of 3. It can be expressed as 19683^{3}, 27^{9}, 3^{27}, and ^{3}3 or when using Knuth's up-arrow notation it can be expressed as and .
- Mathematics: 10^{13} - The approximate number of known non-trivial zeros of the Riemann zeta function as of 2004.^{[5]}
- BioMed — Cells in the human body: the human body consists of roughly 10^{14} cells, of which only 10^{13} are human.^{[6]}^{[7]} The remainder of the cells are bacteria, which mostly reside in the gastrointestinal tract, although the skin is also covered in bacteria.
- BioMed-Insects: 200,000,000,000,000 (2 × 10^{14}) - The estimated number of ants on Earth.
- Computing - MAC-48: 281,474,976,710,656 (2^{48}) possible unique physical addresses.
- Mathematics: 953,467,954,114,363 is the largest known Motzkin prime.
10^{15}
(1000000000000000; 1000^{5}; short scale: one quadrillion; long scale: one thousand billion, or one billiard)
ISO: peta- (P)
- Mathematics: 48,988,659,276,962,496 is the fifth taxicab number.
- Cryptography: There are 7.205759 × 10^{16} different possible keys in the obsolete 56 bit DES symmetric cipher.
10^{18}
(1000000000000000000; 1000^{6}; short scale: one quintillion; long scale: one trillion)
ISO: exa- (E)
- BioMed — Insects: It has been estimated that the insect population of the Earth is about 10^{18}.
- Computing — Manufacturing: An estimated 6 × 10^{18} transistors were produced worldwide in 2008.^{[8]}
- Computing — Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22 × 10^{18}) is equal to 2^{63}-1, and as such is the largest number which can fit into a signed (two's complement) 64-bit integer on a computer.
- Mathematics — NCAA Basketball Tournament: There are 9,223,372,036,854,775,808 (2^{63}) possible ways to enter the bracket.
- Mathematics — Rubik's Cube: There are 43,252,003,274,489,856,000 (about 43 × 10^{18}) different positions of a 3x3x3 Rubik's Cube
- Password strength: Usage of the 95-character set found on standard computer keyboards for a 10-character password yields a computationally intractable 59,873,693,923,837,890,625 (95^{10}, approximately 6 × 10^{19}) permutations.
- Economics: Hyperinflation in Zimbabwe estimated in February 2009 by some economists at 10 sextillion percent,^{[9]} or a factor of 10^{20}
10^{21}
(1000000000000000000000; 1000^{7}; short scale: one sextillion; long scale: one thousand trillion, or one trilliard)
ISO: zetta- (Z)
- Geo — Grains of sand: all the world's beaches put together have been estimated to hold roughly 10^{21} grains of sand.^{[10]}
- Mathematics — Sudoku: There are 6,670,903,752,021,072,936,960 (≈6.7 × 10^{21}) 9×9 sudoku grids.^{[11]}
- Astronomy — Stars: 70 sextillion = 7 × 10^{22} estimated number of stars within range of telescopes (as of 2003), see mass of the observable universe.^{[12]}
- Mathematics: 146,361,946,186,458,562,560,000 (≈1.5 × 10^{23}) is the fifth unitary perfect number.
- Chemistry: there are roughly 6.022 × 10^{23} molecules in one mole of any substance (Avogadro's number)
10^{24}
(1000000000000000000000000; 1000^{8}; short scale: one septillion; long scale: one quadrillion)
ISO: yotta- (Y)
- Mathematics: 2,833,419,889,721,787,128,217,599 (≈2.8 × 10^{24}) is a Woodall prime.
10^{27}
(1000000000000000000000000000; 1000^{9}; short scale: one octillion; long scale: one thousand quadrillion, or one quadrilliard)
- BioMed — Atoms in the human body: the average human body contains roughly 7 × 10^{27} atoms, see ^{[13]}
- Mathematics — Poker: the number of unique combinations of hands and shared cards in a 10-player game of Texas Hold'em is approximately 2.117 × 10^{28}, see Poker probability (Texas hold 'em).
10^{30}
(1000000000000000000000000000000; 1000^{10}; short scale: one nonillion; long scale: one quintillion)
- BioMed: number of bacterial cells on Earth
- Mathematics: The partition of 1000 is 24,061,467,864,032,622,473,692,149,727,991.^{[14]}
- Mathematics: 2^{108} = 324,518,553,658,426,726,783,156,020,576,256 is the last known power of two not containing a digit '9'.^{[15]}
10^{33}
(1000000000000000000000000000000000; 1000^{11}; short scale: one decillion; long scale: one thousand quintillion, or one quintilliard)
- Mathematics: 1,298,074,214,633,706,835,075,030,044,377,087 (≈1.3 × 10^{33}) is a Carol prime
10^{36}
(1000000000000000000000000000000000000; 1000^{12}; short scale: one undecillion; long scale: one sextillion)
- Physics: k_{e}e^{2} / Gm^{2}, the ratio of the electrical to the gravitational forces between two protons, is roughly 10^{36}.
- Computing: The address range of IPv6 (2^{128}) is approximately equal to 3.40282367 × 10^{38}, and is the theoretical maximum number of Internet addresses that can be allocated under the IPv6 addressing system, one more than the largest value that can be represented by a single-precision IEEE floating-point value, the total number of different Universally Unique Identifiers (UUIDs) that can be generated, and in Cryptography, the total number of different possible keys in the AES 128 bit keyspace (symmetric cipher).
10^{39}
(1000000000000000000000000000000000000; 1000^{13}; short scale: one duodecillion; long scale: one thousand sextillion, or one sextilliard)
- Mathematics: 170,141,183,460,469,231,731,687,303,715,884,105,727 (≈1.7 × 10^{38}) is a double Mersenne prime
- Cosmology: The Eddington-Dirac number is roughly 10^{40}.
10^{42} to 10^{100}
(1 000 000 000 000 000 000 000 000 000 000 000 000 000 000; 1000^{14}; short scale: one tredecillion; long scale: one septillion)
- Mathematics: 141×2^{141}+1 = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833 (≈3.9 × 10^{44}) is the second Cullen prime
- Mathematics: There are 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (≈7.401 × 10^{45}) possible permutations for the Rubik's Revenge (4x4x4 Rubik's Cube).
- Chess: 1 × 10^{50} is an estimate of the number of legal chess positions.^{[16]}
- Mathematics: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (≈8 × 10^{53}) is order of Monster group
- Cryptography: There are 6.27710174 × 10^{57} different possible keys in the AES 192 bit keyspace (symmetric cipher).
- Cosmology: 8 × 10^{60} is roughly the number of Planck time intervals since the universe is theorised to have been created in the Big Bang 13.7 ± 0.2 billion years ago
- Mathematics: 4,444,349,792,156,709,907,895,752,551,798,631,908,946,180,608,768,737,946,280,238,078,881 (≈4.4 × 10^{66}) - The largest known prime factor found by ECM factorization as of 2009.^{[17]}
- Mathematics — Cards: 52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 (≈8 × 10^{67}) - the number of ways to order the cards in a 52-card deck.
- Mathematics: There are 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 (2.8287 × 10^{74}) possible permutations for the Professor's Cube (5x5x5 Rubik's Cube).
- Cryptography: There are 1.15792089 × 10^{77} different possible keys in the AES 256 bit keyspace (symmetric cipher).
- Cosmology: various sources estimate the total number of fundamental particles in the observable universe in the range 10^{80} to 10^{85}. However, these estimates are generally regarded as guesswork.
- Mathematics: 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000; 10^{100}, a googol
Larger than 10^{100}
- Board games: 4.8231 × 10^{115}, number of ways to arrange the tiles in English Scrabble (100! / 9! / 2! / 2! / 4! / 12! / 2! / 3! / 2! / 9! / 1! / 1! / 4! / 2! / 6! / 8! / 2! / 1! / 6! / 4! / 6! / 4! / 2! / 2! / 1! / 2! / 1! / 2!).
- Mathematics: There are 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000 000 000 000 000 000 000 (1.5715 × 10^{116}) distinguishable permutations of the V-Cube 6 (6x6x6 Rubik's Cube).
- Chess: Shannon number, 10^{120}, an estimation of the game-tree complexity of chess.
- Physics: 8 × 10^{120}, ratio of the mass-energy in the observable universe to the energy of a photon with a wavelength the size of the observable universe.
- Mathematics — History: Asankhyeya is equal to 10^{140} in ancient India.
- Xiangqi: 10^{150}, an estimation of the game-tree complexity of xiangqi.
- Mathematics: There are 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 (1.9501 × 10^{160}) distinguishable permutations of the V-Cube 7 (7x7x7 Rubik's Cube).
- Physics: 4 × 10^{185}, approximate number of Planck volumes in the observable universe.
- Computing: 1.797 693 134 862 3157 × 10^{308} is approximately equal to the largest value that can be represented by a double-precision IEEE floating-point number.
- Go: 10^{365}, an estimation of the game-tree complexity in the game of Go.^{[citation needed]}
- Mathematics: 2638^{4405} + 4405^{2638} is a 15,071-digit Leyland prime; the largest which has been proven as of 2009.
- Mathematics: 620,366,307,356,565 × 2^{253,824}−1 is a 76424-digit Sophie Germain prime; the largest known as of November 2009.
- Mathematics: 65,516,468,355 · 2^{333,333} ± 1 are 100,355-digit twin primes; the largest known as of 2009.
- Mathematics: 34,790! – 1 is a 142,891-digit factorial prime; the largest known as of 2009.
- Mathematics: 392,113# + 1 is a 169,966-digit primorial prime; the largest known as of 2009.
- Mathematics: 10^{180,004} + 248,797,842×10^{89,998} + 1 is a 180,005-digit palindromic prime, the largest known as of 2008.
- Mathematics: approximately 7.76 · 10^{206,544} cattle in the smallest herd which satisfies the conditions of the Archimedes' cattle problem.
- Mathematics: 24,518^{262,144} + 1 is a 1,150,678-digit Generalized Fermat prime, the largest known as of 2009.
- Mathematics: 19,249 × 2^{13,018,586} + 1 is a 3,918,990-digit Proth prime, the largest known as of 2009.
- Mathematics: 2^{43,112,609} − 1 is a 12,978,189-digit Mersenne prime; the largest known prime of any kind as of 2009.
- Mathematics: 2^{43,112,608} × (2^{43,112,609} − 1) is a 25,956,377-digit perfect number, the largest known as of 2009.
- Mathematics — History: 10^{80,000,000,000,000,000}, largest named number in Archimedes' Sand Reckoner.
- Mathematics: 10^{googol} (), a googolplex.
- Mathematics: , order of magnitude of an upper bound that occurred in a proof of Skewes (later estimated to actually be near 1.397 × 10^{316}).
- Mathematics: , order of magnitude of another upper bound in a proof of Skewes.
- Mathematics: Moser's number "10 in a square" is equal to 10^{10}...,with a total of 1024 layers
- Mathematics: Graham's number, the last ten digits of which are ...24641 95387. Arises as an upper bound solution to a problem in Ramsey theory and is probably the largest number seriously used in a mathematical proof. Representation in powers of 10 would be impractical (the number of digits in the exponent far exceeds the number of particles in the observable universe).
See also
References
- ^ Bridge hands
- ^ The On-Line Encyclopedia of Integer Sequences
- ^ Earth microbes on the Moon
- ^ "Earth microbes on the moon". Science@Nasa. http://science.nasa.gov/newhome/headlines/ast01sep98_1.htm. Retrieved 7 July 2008.
- ^ http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html
- ^ Savage, D. C. (1977), "Microbial Ecology of the Gastrointestinal Tract", Annual Review of Microbiology 31: 107, doi:10.1146/annurev.mi.31.100177.000543
- ^ Berg, R. (1996), "The indigenous gastrointestinal microflora", Trends in Microbiology 4: 430, doi:10.1016/0966-842X(96)10057-3
- ^ http://www.sia-online.org/cs/papers_publications/press_release_detail?pressrelease.id=96
- ^ "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. http://news.bbc.co.uk/1/hi/world/africa/7906031.stm. Retrieved 14 March 2009.
- ^ To see the Universe in a Grain of Taranaki Sand
- ^ Sudoku enumeration
- ^ http://www.rednova.com/news/stories/1/2003/07/22/story004.html
- ^ How many atoms are in the human body?
- ^ (sequence A070177 in OEIS)
- ^ (sequence A035064 in OEIS)
- ^ Victor Allis (1994) (PDF). Searching for Solutions in Games and Artificial Intelligence. Ph.D. Thesis, University of Limburg, Maastricht, The Netherlands. ISBN 9090074880. http://fragrieu.free.fr/SearchingForSolutions.pdf. Retrieved 2010-02-09.
- ^ Paul Zimmermann, "50 largest factors found by ECM".
External links