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On integers which are not differences of two powers

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

Florian Luca
Florian Luca
Affiliation:
Mathematical Institute, UNAM, Campus Morelia, Ap. Postal 61-3 (Xangari). CP 58 089. Morelia. Michoacan. Mexico. E-mail: fluca@matmor.unam.mx
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Extract

Erdős (see [4]) asked if there are infinitely many integers k which are not a difference or a sum of two powers, i.e., if there are infinitely many positive integers k with k≠|um ± vn| for u, v, m, n ε ℤ. This is certainly a very difficult problem. For example, it is known that the Catalan equation, i.e., the equation um − vn = 1 with uv≠0 and min (m, n)≥2 has only finitely many solutions (u, v, m, n), but there is no other positive integer k≥1 for which it is known that the equation

has only finitely many solution (u, v, m, n) with min (m, n)≥2.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 231 - 243
Copyright
Copyright © University College London 2002

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References

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