Consecutive composite values in polynomial sequences

Existing methods for finding long gaps between consecutive primes, equivalently finding long strings of consecutive composite integers, are all based on locating long gaps in the sequence of integers coprime to $P(x)$, the product of primes up to $x$. It is difficult, however, to port these method to related problems, such as the problem of finding many consecutive values of $n$ for which $n^{2}+1$ is composite. The difficulty stems from the fact that a crucial ingredient, a bound for smooth numbers which is the “big tool” in the prime gaps methods, cannot be used for the $n^{2}+1$ problem. In this talk, we review the methods for finding large gaps between primes, and outline a new probabilistic method for proving the existence of long strings of consecutive values of n for which $n^{2}+1$ is composite; that is, strings of $n\leqslant X$ whose length is of order larger than the trivial bound $\log X$. We also discuss the application of our methods to other, related questions. This is joint work with Sergei Konyagin, James Maynard, Carl Pomerance, and Terence Tao.