Numerical results on the transcendence of constants involving , and Euler's constant

Author:
David H. Bailey

Journal:
Math. Comp. **50** (1988), 275-281

MSC:
Primary 11J81; Secondary 11Y60

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917835-1

MathSciNet review:
917835

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a vector of real numbers. *x* is said to possess an integer relation if there exist integers such that . 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 , *e*, and 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 or less: , , , , , , , and . Stronger results were obtained in several cases. These computations thus lend credence to the conjecture that each of the above mathematical constants is transcendental.

**[1]**David H. Bailey,*The computation of 𝜋 to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm*, Math. Comp.**50**(1988), no. 181, 283–296. MR**917836**, https://doi.org/10.1090/S0025-5718-1988-0917836-3**[2]**D. H. Bailey, "A high performance fast Fourier transform algorithm for the Cray-2,"*J. Supercomputing*, v. 1, 1987, pp. 43-60.**[3]**Alan Baker,*Transcendental number theory*, Cambridge University Press, London-New York, 1975. MR**0422171****[4]**J. M. Borwein and P. B. Borwein,*The arithmetic-geometric mean and fast computation of elementary functions*, SIAM Rev.**26**(1984), no. 3, 351–366. MR**750454**, https://doi.org/10.1137/1026073**[5]**J. M. Borwein & P. B. Borwein,*Pi and the AGM--A Study in Analytic Number Theory and Computational Complexity*, Wiley, New York, 1987.**[6]**E. O. Brigham,*The Fast Fourier Transform*, Prentice-Hall, Englewood Cliffs, N. J., 1974.**[7]**H. R. P. Ferguson and R. W. Forcade,*Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two*, Bull. Amer. Math. Soc. (N.S.)**1**(1979), no. 6, 912–914. MR**546316**, https://doi.org/10.1090/S0273-0979-1979-14691-3**[8]**Helaman R. P. Ferguson,*A noninductive 𝐺𝐿(𝑛,𝑍) algorithm that constructs integral linear relations for 𝑛𝑍-linearly dependent real numbers*, J. Algorithms**8**(1987), no. 1, 131–145. MR**875331**, https://doi.org/10.1016/0196-6774(87)90033-2**[9]**Helaman Ferguson,*A short proof of the existence of vector Euclidean algorithms*, Proc. Amer. Math. Soc.**97**(1986), no. 1, 8–10. MR**831375**, https://doi.org/10.1090/S0002-9939-1986-0831375-X**[10]**Dura W. Sweeney,*On the computation of Euler’s constant*, Math. Comp.**17**(1963), 170–178. MR**160308**, https://doi.org/10.1090/S0025-5718-1963-0160308-X

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0917835-1

Article copyright:
© Copyright 1988
American Mathematical Society