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

Abstract

Abstract. Let p be a prime congruent to 1 modulo 4, and let t, u be rational integers such that (t + u √ p )/2 is the fundamental unit of the real quadratic field Q( √ 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 Bn denotes the nth 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 .

A Note on Units of Real Quadratic Fields

“…It is checked that Ankeny-Artin-Chowla conjecture is true for all primes p < 2 × 10 11 in [6], [7] and Mordell conjecture is true for all primes p < 10 7 in [1]. In relation to three consecutive powerful numbers, Mollin and Walsh [5] Let d be a non-square positive integer congruent to 1 modulo 4.…”

An Adaptation of the NICE Cryptosystem to Real Quadratic Orders

“…An ideal p ∈ ker(Φ) as required in encryption step 4 can be determined during encryption or as part of the public key as follows. Generate a random element x ∈ I, with I as given in (6.4), and use Algorithm 5.2 of [14] to find the Z-basis of a reduced principal O Δq -ideal p that has a generator α ∈ O Δq with log 2 (α) ≈ x/ log(2), so log(α) ≈ x. This algorithm is essentially repeated squaring using giant steps and requires O(log(x) log(Δ q )…”

A fast, rigorous technique for computing the regulator of a real quadratic field

“…The information from this section can be found in greater detail in Williams and Wunderlich [26] and in van der Poorten et al [22].…”

“…If we start with a reduced ideal a 1 (so that the cycle starts with the first ideal) that is ambiguous, i.e., such that a 1 = a 1 , we have a special symmetry property for the ideals (see [22]) in the cycle that says that a i = a π+2−i for all i with 1 ≤ i ≤ π, where π is the length of the period. This turns out to be very useful later on.…”