Difference sets and Polynomials of prime variables

Abstract

Let ψ(x) be a polynomial with rational coefficients. Suppose that ψhas the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a prime p such that x-y=ψ(p-1). Furthermore, if P be a set of primes with the positive relative upper density, then there exist x,y\in P and a prime p such that x-y=ψ(p-1).