Editing Regular prime (section)

== Siegel's conjecture ==
It has been [[conjecture]]d that there are [[Infinite set|infinitely]] many regular primes. More precisely {{harvs|first=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1964|txt}} conjectured  that ''[[e (mathematical constant)|e]]''<sup>−1/2</sup>, or about 60.65%, of all prime numbers are regular, in the [[Asymptotic analysis|asymptotic]] sense of [[natural density]]. 

Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers <math>B_k</math>, <math>k=2,\dots,p-3</math>, is not divisible by the prime <math>p</math> is

:<math>\dfrac{p-1}{p}</math>

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime <math>p</math> is

:<math>\left(\dfrac{p-1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}</math>.

By [[E (mathematical constant)#Definitions|the definition of ''e'']], we have

:<math>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{p}=\dfrac{1}{e}</math>

so that we obtain the probability

:<math>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}=e^{-1/2}\approx0.606531</math>.

It follows that about <math>60.6531\%</math> of the primes are regular by chance. Hart et al.<ref>[https://arxiv.org/abs/1605.02398 Irregular primes to two billion, William Hart, David Harvey and Wilson Ong,9 May 2016, arXiv:1605.02398v1]</ref> indicate that <math>60.6590\%</math> of the primes less than <math>2^{31}=2,147,483,648</math> are regular.