Consecutive primes and IP sets

Abstract

For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any number m such that $p_{n+1}-p_n=m$ for infinitely many n. In this note, we show that every IP set of even natural numbers contains infinitely many de Polignac numbers.