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On a conjecture of Erdos and Stewart


Author: Florian Luca
Journal: Math. Comp. 70 (2001), 893-896
MSC (2000): Primary 11D61
DOI: https://doi.org/10.1090/S0025-5718-00-01178-9
Published electronically: March 8, 2000
MathSciNet review: 1677411
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Abstract:

For any $k\ge1$, let $p_k$ be the $k$th prime number. In this paper, we confirm a conjecture of Erdos 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 [Enhancements On Off] (What's this?)

  • [1] Y. Bugeaud & M. Laurent, Minoration effective de la distance $p$-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311-342. MR 98b:11086
  • [2] P. Erdos & 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.
  • [3] R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1994, Problem A2.MR 96e:11002

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Additional Information

Florian Luca
Affiliation: Mathematical Institute, Czech Academy of Sciences, Zitná 25, 115 67 Praha 1, Czech Republic
Email: luca@math.cas.cz

DOI: https://doi.org/10.1090/S0025-5718-00-01178-9
Keywords: $p$-adic linear forms in two logarithms
Received by editor(s): January 4, 1999
Published electronically: March 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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