Abstract view
Sequences with translates containing many primes


Published:19980301
Printed: Mar 1998
Tom Brown
Peter JauShyong Shiue
X. Y. Yu
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Abstract
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
$\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at
least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some
related results.