Irregular Primes to One Million

Irregular primes to 163 million

We compute all irregular primes less than 163 577 856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the λ-invariant is equal to the index of irregularity.

Cyclotomic Invariants for Primes to One Million

Our recent computation of cyclotomic invariants for primes between 125000 and 150000 was extended to one million. No new phenomena appear. This note is a sequel to our recent report [2] on the computation of certain cyclotomic invariants for primes p between 125000 and 150000. That work was based on the table of irregular primes supplied by Tanner and Wagstaff (see [4]). Meanwhile, the extensio...

Computation of Irregular Primes up to Eight Million (preliminary Report)

We report on a joint project with Joe Buhler, Richard Crandall, Reijo Ernvall, and Tauno Metss ankyll a dealing with the computation of irregular primes and cyclo-tomic invariants for primes between four and eight million. This extends previous computations of Buhler et al. 4]. Our computation of the irregular primes is based on a new approach which has originated in the study of Stickelberger ...

Finding Four Million Large Random Primes

On the counting function of irregular primes

It is well-known that there are infinitely many irregular primes. We prove a quantitative version of this statement, namely the number of such primes p ≤ x is at least (1 + o(1)) log log x/ log log log x as x → ∞. We show that the same conclusion holds for the irregular primes corresponding to the Euler numbers. Under some conditional results from diophantine approximation, the above lower boun...