Numerical Results on Relations Between Fundamental Constants Using a New Algorithm

Abstract

Let $x = ({x_1},{x_2}, \ldots ,{x_n})$ be a vector of real numbers, x is said to possess an integer relation if there exist integers ${a_i}$ not all zero such that ${a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0$. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either find a relation if one exists, or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. Recently, one of us discovered a new relation-finding algorithm that is much more efficient, both in terms of run time and numerical precision. This algorithm has now been implemented on high-speed computers, using multiprecision arithmetic. With the help of these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications. In particular, it is established that none of the following constants satisfies a simple, low-degree polynomial: $\gamma$ (Euler’s constant), $\log \gamma$, $\log \pi$, ${\rho _1}$ (the imaginary part of the first zero of Riemann’s zeta function), $\log {\rho _1}$, $\zeta (3)$ (Riemann’s zeta function evaluated at 3), and $\log \zeta (3)$. Several classes of possible additive and multiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum’s constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.