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Almost-prime k-tuples

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  • UserJames Maynard (Oxford)
  • ClockWednesday 21 November 2012, 16:00-17:00
  • HouseMR11, CMS.

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For $i=1,\dots,k$, let $L_i(n)=a_i n+b_i$ be linear functions with integer coefficients, such that $\prod_{i=1}k L_i(n)$ has no fixed prime divisor. It is conjectured that there are infinitely many integers $n$ for which all of the $L_i(n)$ ($1\le i \le k$) are simultaneously prime. Unfortunately we appear unable to prove this, but weighted sieves all us to show that there are infinitely many integers $n$ for which $\prod_[i=1}k L_i(n)$ has at most $r_k$ prime factors, for some explicit constant $r_k$ depending only on $k$. We describe new weighted sieves which improve these bounds when $k\ge 3$, and discuss potential applications to small prime gaps.

This talk is part of the Discrete Analysis Seminar series.

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