### 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 ≤ log_{2}n. An estimate is proved for the number of such n ≤ N.

Original language | English (US) |
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Pages (from-to) | 64-79 |

Number of pages | 16 |

Journal | Journal of Number Theory |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1973 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Journal of Number Theory*, vol. 5, no. 1, pp. 64-79. https://doi.org/10.1016/0022-314X(73)90059-0

**Some applications of Montgomery's sieve.** / Vaughan, R. C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Some applications of Montgomery's sieve

AU - Vaughan, R. C.

PY - 1973/2

Y1 - 1973/2

N2 - 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.

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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UR - http://www.scopus.com/inward/citedby.url?scp=0003109341&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(73)90059-0

DO - 10.1016/0022-314X(73)90059-0

M3 - Article

AN - SCOPUS:0003109341

VL - 5

SP - 64

EP - 79

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -