Hoover's Problem

Abstract

kn + 1, and the integer kn + i = tn! + i, i = 2,..., n 1, is divisible by i and is thus composite. To prove that there exists a largest integer n to which each of its composite totitives can be added to obtain a composite, note first that for n large there exist primes p such that n +p* 0, the inequalities (9) hold with c3 = 1 + e, c4 = 1 E, for all sufficiently large n. This in turn provides the existence of a primep in the interval [n +p*, 2n] such thatp n -O (modp*). Since n <p < 2n, c =p n is a positive composite integer such that c < n and (c, n) = 1. Thus c is a composite totitive of n satisfying n + c =p a prime. The theorem follows since this can be achieved for all large n.