Binary BBP-formulae for logarithms and generalized Gaussian-Mersenne primes

Chamberland, Marc

Journal of integer sequences, Tome 6 (2003) no. 3.

Résumé

Summary: Constants of the form $$ C = \sum_{k=0}^\infty \frac{p(k)}{q(k)b^k} $$ where $p$ and $q$ are integer polynomials, $\deg p \deg q$, and $p(k)/q(k)$ is non-singular for non-negative $k$ and $b\geq 2$, have special properties. The $n$th digit (base $b$) of $C$ may be calculated in (essentially) linear time without computing its preceding digits, and constants of this form are conjectured to be either rational or normal to base $b$. This paper constructs such formulae for constants of the form $\log p$ for many primes $p$. This holds for all Gaussian-Mersenne primes and for a larger class of "generalized Gaussian-Mersenne primes". Finally, connections to Aurifeuillian factorizations are made.

Détail

Classification : 11Y05, 11A41, 11B99, 11T22, 11Y60
Keywords: primes, Gaussian-mersenne, BBP, aurifeuillian