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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On a conjecture of Erdos and Stewart
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by Florian Luca PDF
Math. Comp. 70 (2001), 893-896 Request permission

Abstract:

For any $k\ge 1$, let $p_k$ be the $k$th prime number. In this paper, we confirm a conjecture of Erdős and Stewart concerning all the solutions of the diophantine equation $n!+1=p^a_kp^b_{k+1}$, when $p_{k-1}\le n<p_k$.
References
  • Y. Bugeaud and M. Laurent, Minoration effective de la distance $p$-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), no. 2, 311–342 (French, with English summary). MR 1423057, DOI 10.1006/jnth.1996.0152
  • P. Erdős & R. Obláth, Über diophantische Gleichungen der Form $n!=x^p\pm y^p$ und $n!\pm m!=x^p$, Acta Szeged 8 (1937), 241–255.
  • Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR 1299330, DOI 10.1007/978-1-4899-3585-4
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Additional Information
  • Florian Luca
  • Affiliation: Mathematical Institute, Czech Academy of Sciences, Z̆itná 25, 115 67 Praha 1, Czech Republic
  • MR Author ID: 630217
  • Email: luca@math.cas.cz
  • Received by editor(s): January 4, 1999
  • Published electronically: March 8, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 893-896
  • MSC (2000): Primary 11D61
  • DOI: https://doi.org/10.1090/S0025-5718-00-01178-9
  • MathSciNet review: 1677411