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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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On integers not of the form $\pm p^a\pm q^b$
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Proc. Amer. Math. Soc. 128 (2000), 997-1002 Request permission


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
  • MR Author ID: 254588
  • Email:
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 997-1002
  • MSC (2000): Primary 11B75; Secondary 11B25, 11P32
  • DOI:
  • MathSciNet review: 1695111