Prime divisors of sparse integers (Q1882110)

Let \(g\) and \(s\) be two integers, \(g\leq 2\), \(s\geq 1\). Let \(D\) be \(d_0,\dots, d_s\), a sequence of \(s+ 1\) nonzero integers. Let \(S_{g,s}(D)\) be the set of integers of the form \(d_0+ d_1g^{m_1}+\cdots+ d_s g^{m_s}\) with \(1\leq m_1<\cdots< m_s\). The author proves two theorems: 1. For any fixed \(0<\delta< 1/2\) and every \(s> \max(15,\delta^{-1}- 1)\), when \(X\) is large enough, for \(\sim\pi(X)\) (almost all) primes \(p\leq X\), there exists \(n\in S_{g,s}(D)\), with \(\log n\ll X^{1/2+\delta}\) and such that \(p\) divides \(n\). 2. There exists an absolute constant \(c> 0\) such that for any fixed \(\delta> 0\) and every \(s>\max(4,\delta^{-1}- 1)\), when \(X\) is large enough, for \(\geq c\pi(X)\) primes \(p\leq X\), there exists \(n\in S_{g,s}(D)\), with \(\log n\ll X^{1/2+\delta}\) and such that \(p\) divides \(n\).