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Undecidable arithmetic properties of solutions of Fredholm integral equations.
- Source :
-
Journal of Number Theory . Jul2022, Vol. 236, p230-244. 15p. - Publication Year :
- 2022
-
Abstract
- A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E -functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f (α) is transcendental or algebraic if f (z) is an E -function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f (z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f (0) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6]. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0022314X
- Volume :
- 236
- Database :
- Academic Search Index
- Journal :
- Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 155815473
- Full Text :
- https://doi.org/10.1016/j.jnt.2021.07.018