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.
>On Fri, 17 Dec 1999, Griffith, Shaun wrote:
>> Ian McLoughlin wrote:
>> > Since the list is quiet...
>> > Does a Fibonnacci series contain a finite or an infinite number of
primes?
>> > From what I understand..
>> > In a gen.F sequence if the first two numbers are divisible by a prime
all
>> > its numbers are divisible by the same prime, if the first two numbers
are
>> > co-prime is there a generalised sequence that contains NO PRIMES....
>>
>> The generalized Fibonacci sequence seems to generate at least one prime
>> regardless of the values assigned to Fib(1), Fib(2), *unless* Fib(1) and
>> Fib(2) are even. Then there is never an odd number, and never a chance
for a
>If they are both even they aren't coprime. :)
>
>> prime after Fib(2) (though Fib(1) or Fib(2) may be =2, but that seems
>> trivial).
>>
>> I tried it with composite odd numbers, such as 15,77, which happen to be
>> coprime. The first 3 primes generated are Fib(7)=691, Fib(14)=20101, and
>> Fib(28)=16945081.
>>
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