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Sequences with translates containing many primes

Open Access article
 Printed: Mar 1998
  • Tom Brown
  • Peter Jau-Shyong Shiue
  • X. Y. Yu
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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 $\lambda \leq x^k$ 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.
MSC Classifications: 11A48 show english descriptions unknown classification 11A48 11A48 - unknown classification 11A48

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