A Fibonacci-like Sequence of Composite Numbers

John W. Nicol

Abstract


In 1964, Ronald Graham proved that there exist relatively prime natural numbers $a$ and $b$ such that the sequence $\{A_n\}$ defined by $$ {A}_{n} =A_{n-1}+A_{n-2}\qquad (n\ge 2;A_0=a,A_1=b)$$ 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.


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