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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
Affiliation:
Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I–00146 RomaItalia e-mail: pappa@mat.uniroma3.it, susa@mat.uniroma3.it
ANDREA SUSA
Affiliation:
Dipartimento di Matematica, Università Roma Tre, Largo S. L. Murialdo, 1, I–00146 RomaItalia e-mail: pappa@mat.uniroma3.it, susa@mat.uniroma3.it

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
Copyright
Copyright © Cambridge Philosophical Society 2008

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