# Almost-prime k-tuples

• James Maynard (Oxford)
• Wednesday 21 November 2012, 16:00-17:00
• MR11, CMS.

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.