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On integers not of the form $\pm p^{a}\pm q^{b}$


Author: Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 128 (2000), 997-1002
MSC (2000): Primary 11B75; Secondary 11B25, 11P32
Published electronically: October 27, 1999
MathSciNet review: 1695111
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Abstract: In 1975 F. Cohen and J.L. Selfridge found a 94-digit positive integer which cannot be written as the sum or difference of two prime powers. Following their basic construction and introducing a new method to avoid a bunch of extra congruences, we are able to prove that if

\begin{equation*}\hspace{-1.5pc} x\equiv 47867742232066880047611079\ (\operatorname{mod} 66483034025018711639862527490), \hspace{-1.5pc} \end{equation*}

then $x$ is not of the form $\pm p^{a}\pm q^{b}$ where $p,q$ are primes and $a,b$ are nonnegative integers.


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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@netra.nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05502-1
Received by editor(s): June 16, 1998
Published electronically: October 27, 1999
Additional Notes: This research was supported by the National Natural Science Foundation of the People’s Republic of China and the Return-from-abroad Foundation of the Chinese Educational Committee
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society