Note on the congruence of Ankeny-Artin-Chowla type modulo p²
نویسندگان
چکیده
منابع مشابه
Verification of the Ankeny – Artin – Chowla Conjecture
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...
متن کاملCongruences Related to the Ankeny-artin-chowla Conjecture
Let p be an odd prime with p ⌘ 1 (mod 4) and " = (t + upp)/2 > 1 be the fundamental unit of the real quadratic field K = Q(pp) over the rationals. The Ankeny-Artin-Chowla conjecture asserts that p u, which still remains unsolved. In this paper, we investigate various kinds of congruences equivalent to its negation p | u by making use of Dirichlet’s class number formula, the products of quadrati...
متن کاملComputer verification of the Ankeny-Artin-Chowla Conjecture for all primes less 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 √ 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...
متن کاملOn a Congruence modulo a Prime
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متن کاملCorrigenda and addition to "Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000"
An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes p which are ≡ 5 mod 8, the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field Q(√p) than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than 8p, the main...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1998
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-85-4-377-388