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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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References

[1]Cangelmi, L. and Pappalardi, F.On the r–rank Artin Conjecture, II. J. Number Theory 75 (1999), 120132.CrossRefGoogle Scholar
[2]Hooley, C.On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209220.Google Scholar
[3]Lagarias, J. C. and Odlyzko, A. M.Effective versions of the Chebotarev density theorem. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 409464. Academic Press, London, 1977.Google Scholar
[4]Lang, S.Algebra 2nd edition. Addison-Wesley Publishing Company, Advanced Book Program Reading, MA, 1984. xv+714.Google Scholar
[5]Lenstra, H. W. Jr.On Artin's conjecture and Euclid's algorithm in global fields. Invent. Math. 42 (1977), 201224.CrossRefGoogle Scholar
[6]Matthews, C. R.Counting points modulo p for some finitely generated subgroups of algebraic groups. Bull. London Math. Soc. 14 (1982), 149154.CrossRefGoogle Scholar
[7]Matthews, C. R.A generalisation of Artin's conjecture for primitive roots. Acta Arith. 29 (1976), no. 2, 113146.CrossRefGoogle Scholar
[8]Moree, P.On primes p for which d divides ordp(g). Funct. Approx. Comment. Math. 33 (2005), 8595.Google Scholar
[9]Moree, P. and Stevenhagen, P.A two-variable Artin conjecture. J. Number Theory 85 (2000), no. 2, 291304.CrossRefGoogle Scholar
[10]Murata, L.A problem analogous to Artin's conjecture for primitive roots and its applications. Arch. Math. (Basel) 57 (1991), no. 6, 555565.CrossRefGoogle Scholar
[11]Pappalardi, F.Square free values of the order fuction. New York J. Math. 9 (2003), 331344.Google Scholar
[12]Schinzel, A. and Sierpinski, W.Sur certaines hypothéses concernant les nombres premiers. Acta Arith. 4 (1958) 185208; Erratum ibid. 5(1959), 259.CrossRefGoogle Scholar
[13]Schinzel, A. and Wòjcik, J.On a problem in elementary number theory. Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 2, 225232.CrossRefGoogle Scholar
[14]Wagstaff, S. S. Jr.Pseudoprimes and a generalization of Artin's conjecture. Acta Arith. 41 (1982), no. 2, 141150.CrossRefGoogle Scholar
[15]Wiertelak, K.On the density of some sets of primes p, for which n|ordpa. Funct. Approx. Comment. Math. 28 (2000), 237241.Google Scholar
[16]Wiertelak, K.On the density of some sets of primes p, for which n(ordpb, n)=d. Funct. Approx. Comment. Math. 21 (1992), 6973.Google Scholar
[17]Wójcik, J.On a problem in algebraic number theory. Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 2, 191200.CrossRefGoogle Scholar

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