Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood… Expand

In this brief note, a variety of problems from Ramsey theory on which I would like to see progress made are described and I am offering modest rewards for most of these problems.Expand

We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous… Expand

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation… Expand

A proof of the solution of a century-old problem, known as Schur Number Five, is constructed and validated using a formally verified proof checker, demonstrating that any result by satisfiability solvers can now be validated using highly trustworthy systems.Expand

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large… Expand

For
$$k \in \mathbb {N}$$
, write S(k) for the largest natural number such that there is a k-colouring of
$$\{1, \ldots ,S(k)\}$$
with no monochromatic solution to
$$x-y=z^2$$
. That S(k)… Expand