Computer verification of the Ankeny–Artin–Chowla Conjecture for all primes less than $100000000000$

Abstract:

Let $p$ be a prime congruent to 1 modulo 4, and let $t, u$ be rational integers such that $(t+u\sqrt {p} )/2$ is the fundamental unit of the real quadratic field $\mathbb {Q}(\sqrt {p} )$. The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that $p$ will not divide $u$. This is equivalent to the assertion that $p$ will 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 $p = 40 094 470 441$, then both $t$ and $u$ exceed $10^{330 000}$. In 1988 the AAC conjecture was verified by computer for all $p < 10^{9}$. 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 $p$ up to $10^{11}$.