1999

# Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

## Publication

### Publication

Let $p$ be a prime congruent to 1 modulo~4 and let $t, u$ be rational integers such that $(t+usqrt{p,)/2$ is the fundamental unit of the real quadratic field ${mathbb Q(sqrt{p,)$. The Ankeny-Artin-Chowla Conjecture (AACC) 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 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 $10^{330000$. 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 $10^{11$.

Additional Metadata | |
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Numerical Algorithms and Problems (acm F.2.1) | |

Continued fractions (msc 11A55), Continued fractions and generalizations (msc 11J70), Algebraic number theory computations (msc 11Y40) | |

Life Sciences (theme 5), Energy (theme 4) | |

CWI | |

Modelling, Analysis and Simulation [MAS] | |

Organisation | Scientific Computing |

van der Poorten, A.J, te Riele, H.J.J, & Williams, H.C. (1999).
Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000. Modelling, Analysis and Simulation [MAS]. CWI. |