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On a problem of Schinzel and Wójcik involving equalities between multiplicative orders. (English) Zbl 1242.11070

Summary: Given \(a_1,\ldots , a_r \in \mathbb Q\setminus \{0, \pm 1 \}\), the Schinzel-Wójcik problem is to determine whether there exist infinitely many primes \(p\) for which the order modulo \(p\) of each \(a_1, \ldots, a_r\) coincides. We prove on the GRH that the primes with this property have a density and in the special case when each \(a_i\) 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 \(a_i\)’s are prime, we express the density it terms of an infinite product.

MSC:

11N64 Other results on the distribution of values or the characterization of arithmetic functions
11A07 Congruences; primitive roots; residue systems
11R45 Density theorems
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[1] DOI: 10.1006/jnth.2000.2547 · Zbl 0966.11042 · doi:10.1006/jnth.2000.2547
[2] Murata, Arch. Math. (Basel) 57 pp 555– (1991) · Zbl 0755.11029 · doi:10.1007/BF01199060
[3] Matthews, Acta Arith. 29 pp 113– (1976)
[4] DOI: 10.1112/blms/14.2.149 · Zbl 0484.20019 · doi:10.1112/blms/14.2.149
[5] DOI: 10.1007/BF01389788 · Zbl 0362.12012 · doi:10.1007/BF01389788
[6] DOI: 10.1017/S0305004100074090 · Zbl 0854.11051 · doi:10.1017/S0305004100074090
[7] Wiertelak, Funct. Approx. Comment. Math. 21 pp 69– (1992)
[8] Lagarias, Effective versions of the Chebotarev density theorem. pp 409– (1977) · Zbl 0362.12011
[9] Wiertelak, Funct. Approx. Comment. Math. 28 pp 237– (2000)
[10] Hooley, J. Reine Angew. Math. 225 pp 209– (1967)
[11] Wagstaff, Acta Arith. 41 pp 141– (1982)
[12] DOI: 10.1006/jnth.1998.2319 · Zbl 0926.11086 · doi:10.1006/jnth.1998.2319
[13] DOI: 10.1017/S0305004100070912 · Zbl 0770.11001 · doi:10.1017/S0305004100070912
[14] Schinzel, Acta Arith. 4 pp 185– (1958)
[15] Pappalardi, New York J. Math. 9 pp 331– (2003)
[16] Moree, Funct. Approx. Comment. Math. 33 pp 85– (2005)
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