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On a problem of Schinzel and Wójcik involving equalities between multiplicative orders

Published online by Cambridge University Press:  01 March 2009

FRANCESCO PAPPALARDI  and
ANDREA SUSA
FRANCESCO PAPPALARDI
Affiliation:
Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I–00146 RomaItalia e-mail: [email protected], [email protected]
ANDREA SUSA
Affiliation:
Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I–00146 RomaItalia e-mail: [email protected], [email protected]
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Abstract

Given a1, . . ., ar ∈ ℚ \ {0, ±1}, the Schinzel–Wójcik problem is to determine whether there exist infinitely many primes p for which the order modulo p of each a1, . . ., ar coincides. We prove on the GRH that the primes with this property have a density and in the special case when each ai is a power of a fixed rational number, we show unconditionally that such a density is non zero. Finally, in the case when all the ai's are prime, we express the density it terms of an infinite product.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 303 - 319
Copyright
Copyright © Cambridge Philosophical Society 2008

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