On some applications of finitely generated semi-groups

Shparlinski, Igor E.

doi:10.1007/3-540-58691-1_66

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Finitely Generated Subgroups1

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1996

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“…Estimates of |1 p | have many applications. I. Shparlinski gives an account of some of these in [16]. He notices that argument of Lemma 1.…”

Section: Finitely Generated Subgroupsmentioning

confidence: 95%

On the Order of Finitely Generated Subgroups of Q*(modp) and Divisors ofp−1

Pappalardi

1996

Journal of Number Theory

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Let 1 be a finitely generated subgroup of Q* with rank r. We study the size of the order |1 p | of 1 mod p for density-one sets of primes. Using a result on the scarcity of primes p x for which p&1 has a divisor in an interval of the type [ y, y exp log { y] ({t0.15), we deduce that |1 p | p rÂ(r+1) exp log { p for almost all p and, assuming the Generalized Riemann Hypothesis, we show that |1 p | pÂ ( p) ( Ä ) for almost all p. We also apply this to the Brown Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots. AcademicPress, Inc.

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“…Estimates of |1 p | have many applications. I. Shparlinski gives an account of some of these in [16]. He notices that argument of Lemma 1.…”

Section: Finitely Generated Subgroupsmentioning

confidence: 95%