A Fibonacci-likeSequence of Composite Numbers DONALD E. KNUTH* Stanford University Stanford, CA 94305 Ronald L. Graham [1] found relatively prime integers a and b such that the sequence (AO, A1, A2, . ..) defined by Ao==a, A1 = b, An = Anil+ An-2 (1) contains no prime numbers. His original method proved that the integers a = 331635635998274737472200656430763 b = 1510028911088401971189590305498785 (2) have this property. The purpose of the present note is to show that the smaller pair of integers a = 62638280004239857 b = 49463435743205655 (3) also defines such a sequence. Let (FO, F1, F2,... > be the Fibonacci sequence, defined by (1) with a = 0 and b = 1; and let F_= 1. Then An = Fn-l a + Fnb. (4) Graham's idea was to find eighteen triples of numbers (Pk, Mk, rk) with the properties that * Pk is prime; * Fi is divisible by Pk iff n is divisible by mk; * every integer n is congruent to rk modulo Mk for some k. He chose a and b so that a Fmk-rk bFk-rk+l (mod Pk) (5) It followed that An - O(mod pk) > n rk (mod Mk) (6) for all n and k. Each An was consequently divisible by some Pk; it could not be prime. The eighteen triples in Graham'sconstruction were *This research and/or preparation was supported in part by the National Science Foundation under grant CCR-86-10181, and by the Office of Naval Research contract N00014-87-K-0502. 21 Mathematical Association of America is collaborating with JSTOR to digitize, preserve, and extend access to Mathematics Magazine www.jstor.org ®