Constant Problem

Given an expression involving known constants, integration in finite terms, computation of limits, etc., the constant problem is the determination of if the expression is equal to zero. The constant problem, sometimes also called the identity problem (Richardson 1968) is a very difficult unsolved problem in transcendental number theory. However, it is known that the problem is undecidable if the expression involves oscillatory functions such as sine. However, the Ferguson-Forcade algorithm is a practical algorithm for determining if there exist integers a_i for given real numbers x_i such that


or else establishing bounds within which no relation can exist (Bailey 1988).

See also

Almost Integer, Almost Zero, Ferguson-Forcade Algorithm, Hermite-Lindemann Theorem, Hidden Zero, Identically Zero, Integer Relation, Integration Problem, Richardson's Theorem, Schanuel's Conjecture, Uniformity Conjecture

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Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving pi, e, and Euler's Constant." Math. Comput. 50, 275-281, 1988.Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106, 440-448, 1999.Chen, Z.-Z. and Kao, M.-Y. "Reducing Randomness via Irrational Numbers." 7 Jul 1999., D. "Some Unsolvable Problems Involving Elementary Functions of a Real Variable." J. Symbolic Logic 33, 514-520, 1968.Richardson, D. "The Elementary Constant Problem." In Proc. Internat. Symp. on Symbolic and Algebraic Computation, Berkeley, July 27-29, 1992 (Ed. P. S. Wang). ACM Press, 1992.Richardson, D. "How to Recognize Zero." J. Symb. Comp. 24, 627-645, 1997.Sackell, J. "Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations." Trans. Amer. Math. Soc. 336, 151-171, 1993.

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Constant Problem

Cite this as:

Weisstein, Eric W. "Constant Problem." From MathWorld--A Wolfram Web Resource.

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