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Mathematics of Computation

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

Author: David H. Bailey
Journal: Math. Comp. 50 (1988), 275-281
MSC: Primary 11J81; Secondary 11Y60
MathSciNet review: 917835
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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.

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Article copyright: © Copyright 1988 American Mathematical Society