A. J. van der Poorten, H. J. J. te Riele, Hywel C Williams,
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 result that the AAC conjecture is true for all the primes ≡ 1 mod 4 which are < 10 11 , remains valid.As an addition, ...
Tópico(s): Analytic Number Theory Research
2002 - American Mathematical Society | Mathematics of Computation
A. J. van der Poorten, H. J. J. te Riele, Hywel C Williams,
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 ...
Tópico(s): Advanced Mathematical Theories and Applications
2000 - American Mathematical Society | Mathematics of Computation
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Tópico(s): Railway Systems and Energy Efficiency
1970 - WIT Press | WIT transactions on the built environment