Erdős–Kac theorem

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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, loosely speaking, the probability distribution of

 \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}}

is the standard normal distribution. This is a deep extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size \sqrt{\log\log n}.

More precisely, for any fixed a < b,

\lim_{x \rightarrow \infty}  \left ( \frac {1}{x} \cdot \#\left\{ n \leq x : a \le \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \le b \right\} \right ) = \Phi(a,b)

where \Phi(a,b) is the normal (or "Gaussian") distribution, defined as

\Phi(a,b)= \frac{1}{\sqrt{2\pi}}\int_a^b e^{-t^2/2} \, dt.

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 log log n.

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



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
1065 66 5 2.2
109,566 9,567 10 3.2
10210,704,568 210,704,569 20 4.5
101022 1022+1 50 7.1
101044 1044+1 100 10
1010434 10434+1 1000 31.6
A spreading Gaussian distribution of distinct primes illustrating the Erdos-Kac theorem

Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% (±σ) are constructed from between 7 and 13 primes.

A hollow sphere the size of the planet Earth filled with fine sand would have around 1033 grains. A volume the size of the observable universe would have around 1093 grains of sand. There might be room for 10185 quantum strings in such a universe.

Numbers of this magnitude—with 186 digits—would require on average only 6 primes for construction.


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