Hugh Williams - Computer verification of the Ankeny-Artin-Chowla conjecture for all p< 5.10¹⁰

Let p be a prime congruent to 1 modulo 4 and let t, u be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture ( AAC conjecture) asserts that pwill not divide u. This is equivalent to the assertion that pwill not divide B_(p-1)/2, where B_n denotes the n-th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers t, u; for example, when p = 40094470441, then both t and u exceed 10³³⁰⁰⁰⁰. In 1988 the AACconjecture was verified by computer for all p<10⁹. In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes pup to 5.10¹⁰.

This is joint work with Alf van der Poorten and Herman te Riele.