A Fibonacci-like Sequence of Composite Numbers

In 1964, Ronald Graham proved that there exist relatively prime natural numbers a and b such that the sequence {An} defined by An = An-1 + An-2 (n 2; A0 = a, A1 = b) contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth's computation, Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham's construction and generalize Wilf's note, and show that the 12-digit pair (a, b) = (407389224418, 76343678551) also defines such a sequence. Mathematical Reviews Subject Numbers: 11B39, 11N99.

• Type: Journal
• Year: 1999
• Where: COMBINATORICS
• Authors: John W. Nicol
• Keywords: 17-digit Pair, COMBINATORICS 1999, Prime Natural Numbers, Ronald Graham