|HUGH WILLIAMS, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada|
|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 Bn 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.