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finding irregular primes below a limit with PARI/gp

Irregular primes were defined/known by Kummer, around 1850.  Kummer studied cyclotomic extensions of Q and, I think, their ring of integers (in modern parlance).

The series of commands below has PARI/gp find the irregular primes (the number of them) below 125,000;  this and more was done in a paper of Wagstaff in 1978.  He found:

3559  irregular primes of index 1,

875   irregular primes of index 2,

153  irregular primes of index 3,

16   irregular primes of index 4  and

2     irregular primes of index 5.

3559+875+153+16+2
 = 4605    irregular primes up to 125,000  (Wagstaff).

Using PARI/gp, Bernoulli numbers (as provided by berfrac()),

k=125000;

g2=primes(k);

np=primepi(k);

w3=vector(np);

for(Z=6,k/2,a=2*Z;b=numerator(abs(bernfrac(a)));for(X=1,np,if(g2[X]>a+2,if(lift(Mod(b,g2[X]))<1,w3[X]=1))));

sum(X=1,np,w3[X])

%126 = 4605     // PARI-gp  output

 

This agrees with Wagstaff’s computation in 1978.  The method above is not the best:  there are congruences known for the Bernoulli numbers.  It took a few days on PARI/gp.

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Written by meditationatae

May 15, 2013 at 12:39 pm

Posted in History

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