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 - 2k 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 | |
| State | Published - Feb 1973 |
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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
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Some applications of Montgomery's sieve. / Vaughan, R. C.
In: Journal of Number Theory, Vol. 5, No. 1, 02.1973, p. 64-79.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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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 -