From Wikipedia, the free encyclopedia
In number
theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as
the fundamental theorem of probabilistic number
theory, states that if ω(n) is the number of distinct
prime factors of
n, then for any fixed
a < b,
where Φ(a,b) is
the normal (or "Gaussian")
distribution, defined as
Stated somewhat heuristically, what Erdős and Kac proved was
that if n is a randomly
chosen large integer, then the number of distinct prime factors of
n has approximately the
normal distribution with mean and variance
for
This means that the construction of a number around one billion
requires on average three primes.
For example 1,000,000,003 =
23 × 307 × 141623.
n 
Number of
Digits in n

Average number
of distinct primes

standard
deviation

1,000 
4 
2 
1.4 
1,000,000,000 
10 
3 
1.7 
1,000,000,000,000,000,000,000,000 
25 
4 
2 
10^{65} 
66 
5 
2.2 
10^{9,566} 
9,567 
10 
3.2 
10^{210,704,568} 
210,704,569 
20 
4.5 
10^{10}22 
10^{22} 
50 
7.1 
10^{10}44 
10^{44} 
100 
10 
10^{10}434 
10^{434} 
1000 
31.6 
References
 Paul Erdős
and Mark Kac, "The
Gaussian Law of Errors in the Theory of Additive Number Theoretic
Functions", American Journal of Mathematics, volume 62,
No. 1/4, (1940), pages 738–742.
External
links