Some Applications of the Maynard–Tao Theorem

Let N and P be the set of natural numbers and prime numbers, respectively. For a positive integer n, define the following two representation functions fN(n)=#{(p,k):n=p+2k, p∈P,k∈N} and fP(n)=#{(p,l):n=p+2l, p,l∈P}.An old result of Erdős showed that fN(n) is unbounded. As an elementary application of the Maynard–Tao theorem, we prove that fP(n) is also unbounded, which confirms a conjecture of Chen in 2010.