Some applications of Montgomery's sieve

R. C. Vaughan

doi:10.1016/0022-314X(73)90059-0

Some applications of Montgomery's sieve

R. C. Vaughan

Mathematics

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

Artin has conjectured that every positive integer not a perfect square is a primitive root for some odd prime. A new estimate is obtained for the number of integers in the interval [M + 1, M + N] which are not primitive roots for any odd prime, improving on a theorem of Gallagher. Erdo{combining double acute accent}s has conjectured that 7, 15, 21, 45, 75, and 105 are the only values of the positive integer n for which n - 2^k is prime for every k with 1 ≤ k ≤ log₂n. An estimate is proved for the number of such n ≤ N.

Original language	English (US)
Pages (from-to)	64-79
Number of pages	16
Journal	Journal of Number Theory
Volume	5
Issue number	1
DOIs	https://doi.org/10.1016/0022-314X(73)90059-0
State	Published - Feb 1973

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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AB - Artin has conjectured that every positive integer not a perfect square is a primitive root for some odd prime. A new estimate is obtained for the number of integers in the interval [M + 1, M + N] which are not primitive roots for any odd prime, improving on a theorem of Gallagher. Erdo{combining double acute accent}s has conjectured that 7, 15, 21, 45, 75, and 105 are the only values of the positive integer n for which n - 2k is prime for every k with 1 ≤ k ≤ log2n. An estimate is proved for the number of such n ≤ N.

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