Numerical results on the transcendence of constants involving $\pi ,e$, and Euler’s constant

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 ${a_i}$ such that ${a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0$. Recently, Ferguson and Forcade discovered practical algorithms [7], [8], [9] which, for any n, either find a relation if one exists or else establish bounds within which no relation can exist. One obvious application of these algorithms is to determine whether or not a given computed real number satisfies any algebraic equation with integer coefficients (where the sizes of the coefficients are within some bound). The recursive form of the Ferguson-Forcade algorithm has been implemented with multiprecision arithmetic on the Cray-2 supercomputer at NASA Ames Research Center. The resulting computer program has been used to probe the question of whether or not certain constants involving $\pi$, e, and $\gamma$ satisfy any simple polynomial equations. These computations established that the following constants cannot satisfy any algebraic equation of degree eight or less with integer coefficients whose Euclidean norm is ${10^9}$ or less: $e/\pi$, $e + \pi$, ${\log _e}\pi$, $\gamma$, ${e^\gamma }$, $\gamma /e$, $\gamma /\pi$, and ${\log _e}\gamma$. Stronger results were obtained in several cases. These computations thus lend credence to the conjecture that each of the above mathematical constants is transcendental.