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A Two-Variable Artin Conjecture

✍ Scribed by Pieter Moree; Peter Stevenhagen

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 145 KB
Volume: 85
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2000.2547

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✦ Synopsis

Let a, b # Q* be rational numbers that are multiplicatively independent. We study the natural density $(a, b) of the set of primes p for which the subgroup of F p * generated by (a mod p) contains (b mod p). It is shown that, under assumption of the generalized Riemann hypothesis, the density $(a, b) exists and equals a positive rational multiple of the universal constant S=> p prime (1& pÂ( p 3 &1)). An explicit value of $(a, b) is given under mild conditions on a and b. This extends and corrects earlier work of Stephens (1976, J. Number Theory 8, 313 332). We also discuss the relevance of the result in the context of second order linear recurrent sequences and some numerical aspects of the determination of $(a, b). 2000

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