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On a distribution property of the residual order of a (mod p)

✍ Scribed by Koji Chinen; Leo Murata

Publisher: Elsevier Science
Year: 2004
Tongue: English
Weight: 286 KB
Volume: 105
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2003.06.005

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✦ Synopsis

Let a be a positive integer with aa1 and Q a ðx; k; lÞ be the set of primes ppx such that the residual order of a in Z=pZ Â is congruent to l mod k: It seems that no one has ever considered the density of Q a ðx; k; lÞ for la0 when kX3: In this paper, the natural densities of Q a ðx; 4; lÞ ðl ¼ 0; 1; 2; 3Þ are considered. We assume a is square free and a 1 ðmod 4Þ: Then, for l ¼ 0; 2; we can prove unconditionally that their natural densities are equal to 1/3. On the contrary, for l ¼ 1; 3; we assume the Generalized Riemann Hypothesis, then we can prove their densities are equal to 1/6.

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On a distribution property of the residual order of a&#xa0;(mod&#xa0;p)

✦ Synopsis

On a distribution property of the residual order of a (mod p)