Computer verification of the Ankeny-Artin-Chowla conjecture for all primesless than 100 000 000 000

Let p be a prime congruent to 1 modulo 4, and let t, u be rational integers such that (t + u rootp)/2 is the fundamental unit of the real quadratic fi eld Q(rootp). The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts t hat p will not divide u. This is equivalent to the assertion that p will no t divide B(p-1)/2, where B-n denotes the nth Bernoulli number. Although fir st 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 th e 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 s ome results of a computer run of the method for all primes p up to 10(11).