TY - JOUR

T1 - Some applications of Montgomery's sieve

AU - Vaughan, R. C.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

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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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 -