Fibonacci-like primefree sequences

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A083216: Fibonacci-like primefree sequence.

{ 20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, ... }

Like the sequence of Fibonacci numbers, this sequence has two initial terms set, and the rest are given by the familiar recurrence relation

a (n) = a (n  − 1) + a (n  − 2), n   ≥   2

. The amazing thing about this sequence is that there are no prime numbers among any of its terms even though the initial terms

a (0) = 2 × 5 × 5623 × 366631232537

and

a (1) = 3 × 1264921787189171

are coprime.

And whereas the search for prime numbers seeks larger and larger primes, the search for primefree sequences seeks smaller initial terms. When in 1964, Ronald Graham (of Graham’s number fame) proved that this kind of sequence is possible, his example had initial terms with more than thirty digits each (in base 10). Donald Knuth found a 17-digit pair in 1990, and later that same year Herbert Wilf found this sequence with slightly smaller initial terms. The current record is a 12-digit pair found by John W. Nicol in 1999.