Numerical results on relations between fundamental constants using a new algorithm

Authors:
David H. Bailey and Helaman R. P. Ferguson

Journal:
Math. Comp. **53** (1989), 649-656

MSC:
Primary 11Y16; Secondary 68Q25

DOI:
https://doi.org/10.1090/S0025-5718-1989-0979934-9

MathSciNet review:
979934

Full-text PDF Free Access

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 not all zero such that . 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: (Euler's constant), , , (the imaginary part of the first zero of Riemann's zeta function), , (Riemann's zeta function evaluated at 3), and . 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.

**[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]**David H. Bailey,*Numerical results on the transcendence of constants involving 𝜋,𝑒, and Euler’s constant*, Math. Comp.**50**(1988), no. 181, 275–281. MR**917835**, https://doi.org/10.1090/S0025-5718-1988-0917835-1**[3]**D. H. Bailey, "A high performance FFT algorithm for vector supercomputers,"*Internat. J. Supercomput Appl.*, v. 2, 1988, pp. 82-87.**[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 Computation Complexity*, Wiley, New York, 1987.**[6]**J. M. Borwein & P. B. Borwein, "Ramanujan and Pi,"*Sci. Amer.*, v. 258, 1988, pp. 112-117.**[7]**J. V. Drazil,*Quantities and Units of Measurement*:*A Dictionary and Handbook*, Mansell Publishing, Ltd., London, 1983.**[8]**Mitchell J. Feigenbaum,*Quantitative universality for a class of nonlinear transformations*, J. Statist. Phys.**19**(1978), no. 1, 25–52. MR**0501179**, https://doi.org/10.1007/BF01020332**[9]**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**[10]**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**[11]**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**[12]**H. R. P. Ferguson,*A New Integral Relation Finding Algorithm Involving Partial Sums of Squares*, Brigham Young University preprint, Sept. 1987.**[13]**H. R. P. Ferguson, "A new integral relation finding algorithm involving partial sums of squares and no square roots,"*Abstracts Amer. Math. Soc.*, v. 9, 1988, p. 214.**[14]**I. J. Good,*On the masses of the proton, neutron and hyperons*, J. Roy. Naval Sci. Service**12**(1957), 144. MR**0090432****[15]**Ravi Kannan and Lyle A. McGeoch,*Basis reduction and evidence for transcendence of certain numbers*, Foundations of software technology and theoretical computer science (New Delhi, 1986) Lecture Notes in Comput. Sci., vol. 241, Springer, Berlin, 1986, pp. 263–269. MR**889925**, https://doi.org/10.1007/3-540-17179-7_16**[16]**J. Håstad, B. Helfrich, J. Lagarias, and C.-P. Schnorr,*Polynomial time algorithms for finding integer relations among real numbers*, STACS 86 (Orsay, 1986) Lecture Notes in Comput. Sci., vol. 210, Springer, Berlin, 1986, pp. 105–118. MR**827729**, https://doi.org/10.1007/3-540-16078-7_69**[17]**A. M. Odlyzko and H. J. J. te Riele,*Disproof of the Mertens conjecture*, J. Reine Angew. Math.**357**(1985), 138–160. MR**783538**, https://doi.org/10.1515/crll.1985.357.138**[18]**B. Robertson, "Wyler's expression for the fine-structure constant ,"*Phys. Rev. Lett.*, v. 27, 1971, pp. 1545-1547.**[19]**Paul N. Swarztrauber,*Multiprocessor FFTs*, Proceedings of the international conference on vector and parallel computing—issues in applied research and development (Loen, 1986), 1987, pp. 197–210. MR**898043**, https://doi.org/10.1016/0167-8191(87)90018-4

Retrieve articles in *Mathematics of Computation*
with MSC:
11Y16,
68Q25

Retrieve articles in all journals with MSC: 11Y16, 68Q25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0979934-9

Article copyright:
© Copyright 1989
American Mathematical Society