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On Minimal Sets of Generators for Primitive Roots

Published online by Cambridge University Press:  20 November 2018

Francesco Pappalardi*
Affiliation:
Dipartimento di Matematica, Terza Università degli Studi di Roma, Via Corrado Segre, 4, Roma 00146-Italia, e-mail: pappa@mat.uniroma3.it
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Abstract

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A conjecture of Brown and Zassenhaus (see [2]) states that the first log/? primes generate a primitive root (mod p) for almost all primes p. As a consequence of a Theorem of Burgess and Elliott (see [3]) it is easy to see that the first log2 p log log4+∊ p primes generate a primitive root (mod p) for almost all primes p. We improve this showing that the first log2 p/ log log p primes generate a primitive root (mod p) for almost all primes p.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Bombieri, E., Le grande crible dans la théorie analytique des nombres, Astérisque 18(1974).Google Scholar
2. Brown, H. and Zassenhaus, H., Some empirical observation on primitive roots, J. Number Theory 3(1971) 306309.Google Scholar
3. Burgess, D. A. and Elliott, P. D. T. A., The average of the least primitive root, Mathematika 15(1968), 3950.Google Scholar
4. Canfield, E. R., Erdös, P. and Pomerance, C., On a problem of Oppenheim concerning “Factorization Numerorum ”, J. Number Theory 17(1983), 128.Google Scholar
5. Konyagin, S. and Pomerance, C., On primes recognizable in deterministic polynomial time, preprint.Google Scholar
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