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Thursday, January 26, 2006

**Abstract:** The techniques that Goldston, Pintz, and Yildirim recently used to prove the existence of short gaps between primes can be applied to other sequences. For example, one can apply these techniques to the sequence of numbers that are products of exactly two primes. Using this, we can prove that there are infinitely many integers n such that at least two of the numbers n, n+2, n+6 are products of exactly two primes. The same can be done for more general linear forms; e.g., there are infintely many n such at least two of 42n+1, 44n+1, 45n+1 are products of exactly two primes. This in turn leads to simple proofs of Heath-Brown's theorem that d(n)=d(n+1) infinitely often and of Schlage-Puchta's theorem that $\omega(n)=\omega(n+1)$ infinitely often. With other choices of linear forms, we can sharpen this to d(n)=d(n+1)=24 and $\omega(n)=\omega(n+1)=3$ infinitely often. This is joint work with D. Goldston, J. Pintz, and C. Yildirim.