Treffer: Integers not of the form c(2a+2b)+pα: Integers not of the form \(c(2^a+2^b)+p^{\alpha}\)

Schinzel observed that \(2^{2^n}-1\) is not of the form \(2^a+2^b+p\), where \(a>b\) are positive integers and \(p\) is a prime. Combining Schinzel's observation with Erdős's method of covering congruences, Crocker showed that there are infinitely many positive integers not of the form \(2^a+2^b+p\) for any positive integers \(a\) and \(b\) and prime \(p\). In 2001, Sun asked whether it is true that for any positive integer \(c\), there exist infinitely many positive integers not of the form \(c(2^a+2^b)+p^{\alpha}\), where \(a,b,c,\alpha\) are natural numbers and \(p\) is prime. In this short and elegant paper, the author confirms this conjecture. His main tools are Crocker's covering congruence system together with Schlickewei's finiteness theorem for the number of non-degenerate solutions to \({\mathcal S}\)-unit equations. The paper concludes with a list of four conjectures, an example of which is that the set of positive integers not of the form \(2^a+p^{\alpha}q^{\beta}\), where \(a,~\alpha,~\beta\) are natural numbers and \(p\) and \(q\) are primes is of asymptotic density zero.