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Mathematics of Computation

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Solution of Nathanson's exponential congruence

Author: Samuel S. Wagstaff
Journal: Math. Comp. 33 (1979), 1097-1100
MSC: Primary 10A10
MathSciNet review: 528063
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Abstract: The exponential congruence $ {5^n} \equiv 2\;\pmod {3^n}$ has no solution $ n > 1$. This result is proved by using a theorem of van der Poorten to produce an upper bound for the size of such solutions n which is within range of machine verification, and then checking that no n below this bound satisfies the congruence.

References [Enhancements On Off] (What's this?)

  • [1] M. B. NATHANSON, "An exponential congruence of Mahler," Amer. Math. Monthly, v. 79, 1972, pp. 55-57. MR 46 #133. MR 0300973 (46:133)
  • [2] A. J. VAN DER POORTEN, "Linear forms in logarithms in the p-adic case," Chapter 2 of Transcendence Theory: Advances and Applications, (A. Baker and D. W. Masser, Eds.), Academic Press, London, 1977. MR 0498418 (58:16544)
  • [3] A. SCHINZEL, "On two theorems of Gelfond and some of their applications," Acta Arith., v. 13, 1967, pp. 177-236. MR 36 #5086. MR 0222034 (36:5086)

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Keywords: Exponential congruence
Article copyright: © Copyright 1979 American Mathematical Society

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