On the Order of Finitely Generated Subgroups of Q*(mod p) and Divisors ofp−1
✍ Scribed by Francesco Pappalardi
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 654 KB
- Volume
- 57
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
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 | p ( p) ( Ä ) for almost all p. We also apply this to the Brown Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots. 1996 Academic Press, Inc. ord p (a)>-pÂlog p (1.1) for all but O(xÂlog 3 x) primes p x.