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On q-orders in primitive modular groups

Jacek Pomykała (2014)

Acta Arithmetica

We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that * p has no generator in the interval [1,B]. As a consequence we prove that if Q > e x p [ c ( l o g p ) / ( l o g B ) ( l o g l o g p ) ] with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that ν q ( o r d p b ) = ν q ( p - 1 ) for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.

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