An old book of mine gives without proof an example of Fibonacci Sequence
that countains no primes, but where U(1) and U(2) are co-prime.
The sequence was found by R. L. Graham.
Reference :
"A Fibonacci-like sequence of composite numbers",
R.L. Graham, Math. Mag. 37, 1964.
U(1) = 1786772701928802632268715130455793
U(2) = 1059683225053915111058165141686995
U(N+2) = U(N+1) + U(N)
I only verified with Mapple that U(1) and U(2) are co-prime
and that U(N) is composite for N<10.
I checked a few thousand terms, and they were all composite.
+--------------------------------------------------------+
|
Jud
McCranie
|
|
|
| 137*2^197783+1 is prime! (59,541 digits,
11/11/99) |
+--------------------------------------------------------+
