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L. Adleman (1980)
On distinguishing prime numbers from composite numbers21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
Math. Z. 243, 79–84 (2003) Mathematische Zeitschrift DOI: 10.1007/s00209-002-0449-z n n An upper bound for the G.C.D. of a − 1 and b − 1 1 2 3 Yann Bugeaud , Pietro Corvaja and Umberto Zannier Universite ´ Louis Pasteur, Mathematiques, ´ 7, rue Rene ´ Descartes, 67084 Strasbourg Cedex, France (e-mail: bugeaud@math.u-strasbg.fr) Dipartimento di Matematica e Inf., Via delle Scienze, 206, 33100 Udine, Italy (e-mail: corvaja@dimi.uniud.it) I.U.A.V. - DCA, S. Croce 191, 30135 Venezia, Italy (e-mail: zannier@iuav.it) Received: 27 April 2001 / Published online: 8 November 2002 – c Springer-Verlag 2002 1 Introduction It is a known amusing elementary problem to prove that, if a − 1 divides b − 1 for every large positive integer n, then b is a power of a. (Here a, b are integers greater than 1.) This is a particular case of the so-called Hadamard Quotient theorem, n n concerning arbitrary recurrent sequences in place of {a − 1} and {b − 1} (see [3]). The general case was proved by A.J. van der Poorten [4] by quite ingenious arguments of p-adic nature and rather complicated auxiliary con- structions. Nevertheless, the simple case under consideration (as well as the cases where
Mathematische Zeitschrift – Springer Journals
Published: Jan 1, 2003
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