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Arithmetic progressions of primitive roots of a prime. III. (English) Zbl 0243.10002

The author continues the study of the distribution of the primitive roots of a prime, using only combinatorial methods. [Parts I and Part II, J. Reine Angew. Math. 235, 185–188 (1969; Zbl 0172.32502); 244, 108–111 (1970; Zbl 0205.34703)].
The principal result of this paper is given in the following theorems:
Theorem 1. Let each of \(r\), \(s\) and \(t\) be a positive integer. there is a positive integer \(N\), depending only on \(r\) and \(s\) such that if \(p>N\) is a prime with \(p-1\) divisible by at most \(s\) distinct primes, then for each primitive root \(g\) modulo \(p^t\) there is an arithmetic progression of primitive roots modulo \(p^t\) with common difference \(g\).
(In part II of this series, a result of this type was proved, but it required the \(s\) distinct primes of the theorem to be the first \(s\) primes.)
Theorem 2. If \(p\equiv 1\pmod 4\) is a prime and \(p-1\) is divisible by at most three distinct primes, then \(p\) has at least one pair prime of consecutive primitive roots. If \(p\equiv 3\pmod 4\) is a prime greater than 7 and \(p-1\) is divisible by at most two distinct primes, then there is at least one pair of consecutive primitive roots modulo \(p\).
{The goal toward which one is directed is to show by combinatorial methods only that if \(s\) is a natural number then for each sufficiently large prime \(p\) there are at least \(s\) consecutive primitive roots modulo \(p\). Using character sum estimates, J. Johnsen [J. Reine Angew. Math. 251, 10–19 (1971; Zbl 0237.10026)] has proved these results. If a specific number of primitive roots is desired then exacatly which primes are excluded? E. g. as in Theorem 2, which primes do not have a pair of consecutive primitive roots?}

MSC:

11B25 Arithmetic progressions
11A07 Congruences; primitive roots; residue systems
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