Sequences with Translates Containing Many Primes
Canadian mathematical bulletin, Tome 41 (1998) no. 1, pp. 15-19
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Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers $k$ and $N$ , there exists a positive integer $\lambda $ such that ${{n}^{k}}\,+\,\lambda $ is prime for at least $N$ positive integers $n$ . In other words, there exists $\lambda $ such that ${{n}^{k}}\,+\,\lambda $ , represents at least $N$ primes.We give a quantitative version of this result.We show that there exists $\lambda \le {{x}^{k}}$ such that ${{n}^{k}}\,+\,\lambda $ , 1 ≤ n ≤ x, represents at least $\left( \frac{1}{k}\,+\,o\left( 1 \right) \right)\,\pi \left( x \right)$ primes, as $x\to \infty $ . We also give some related results.
Brown, Tom; Shiue, Peter Jau-Shyong; Yu, X.Y. Sequences with Translates Containing Many Primes. Canadian mathematical bulletin, Tome 41 (1998) no. 1, pp. 15-19. doi: 10.4153/CMB-1998-003-3
@article{10_4153_CMB_1998_003_3,
author = {Brown, Tom and Shiue, Peter Jau-Shyong and Yu, X.Y.},
title = {Sequences with {Translates} {Containing} {Many} {Primes}},
journal = {Canadian mathematical bulletin},
pages = {15--19},
year = {1998},
volume = {41},
number = {1},
doi = {10.4153/CMB-1998-003-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-003-3/}
}
TY - JOUR AU - Brown, Tom AU - Shiue, Peter Jau-Shyong AU - Yu, X.Y. TI - Sequences with Translates Containing Many Primes JO - Canadian mathematical bulletin PY - 1998 SP - 15 EP - 19 VL - 41 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-003-3/ DO - 10.4153/CMB-1998-003-3 ID - 10_4153_CMB_1998_003_3 ER -
%0 Journal Article %A Brown, Tom %A Shiue, Peter Jau-Shyong %A Yu, X.Y. %T Sequences with Translates Containing Many Primes %J Canadian mathematical bulletin %D 1998 %P 15-19 %V 41 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-003-3/ %R 10.4153/CMB-1998-003-3 %F 10_4153_CMB_1998_003_3
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