# Constant problem

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In mathematics, the **constant problem** is the problem of deciding whether a given expression is equal to zero.

## Contents

## The problem[edit]

This problem is also referred to as the **identity problem**^{[1]} or the method of **zero estimates**. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction. Specifically, they use some auxiliary function to create an integer *n* ≥ 0, which is shown to satisfy *n* < 1. Clearly, this means that *n* must have the value zero, and so a contradiction arises if one can show that in fact *n* is *not* zero.

In many transcendence proofs, proving that *n* ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general methods for attacking it. The number *n* that arises may involve integrals, limits, polynomials, other functions, and determinants of matrices.

## Results[edit]

In certain cases algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if *x*_{1}, ..., *x*_{n} are real numbers, then there is an algorithm^{[2]} for deciding whether there are integers *a*_{1}, ..., *a*_{n} such that

If the expression we are interested in contains a oscillating function, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.

## See also[edit]

## References[edit]

**^**Richardson, Daniel (1968). "Some Unsolvable Problems Involving Elementary Functions of a Real Variable".*Journal of Symbolic Logic*.**33**: 514–520. doi:10.2307/2271358.**^**Bailey, David H. (January 1988). "Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant" (PDF).*Mathematics of Computation*.**50**(20): 275–281. doi:10.1090/S0025-5718-1988-0917835-1.