On the convergence of series of reciprocals of primes related to the Fermat numbers
For any integer \(m\geq 0\) let \(F_m\) be the \(m\)th Fermat number. Let us denote by \(P\) the set of all primes \(p\) for which there exists \(m\) such that \(p\) divides \(F_m\). In 1955, \textit{S. W. Golomb} [Math. Scand. 3, 264-274 (1955; Zbl 0067.27503)] asked if the series whose general term is \(1/p\) \((p\in P)\) is convergent. A positive answer is given in this paper. \textit{A. Aigner} [Monatsh. Math. 101, 85-93 (1986; Zbl 0584.10003)] introduced the notion of an elite prime as being a prime number \(p\) such that \(p\) is a quadratic nonresidue modulo \(F_m\) for all but finitely many values of \(m\). Denote by \(E\) the set of all elite primes. It is shown that the series whose general term is \(1/p\) with \(p\in E\) is convergent.
scientific article; zbMATH DE number 1891430