On integers 2(p + ia) not of the form $${a^k + \phi(m)}$$ a k + ϕ ( m )

Abstract

Let \({a \geqq 2}\) be an even integer and let \({L \geqq 1}\) be an integer. We show that for a sufficiently large x, the number of primes \({p \leqq x}\) such that 2p + 2a, . . . , 2p + 2La can not be expressed as \({a^{k} +\phi(m)}\) is at least \({C(a,L) \frac{x}{log x}}\), where k, m are positive integers, \({\phi(m)}\) is the Euler totient function and the constant C(a, L) depends on a, L.