[seqfan] Re: More (composite) terms for A233281
Hans Havermann
gladhobo at teksavvy.com
Wed Feb 12 23:03:22 CET 2014
113 = 6*19-1: I think Fib(p) has prime factors 2*p*n - 1 (n odd, not ending in 5), 2*p*n + 1 (n even).
The smallest prime factor of Fib(p) being ~ 2p suggests that the smallest composite divisor of Fib(p) exceeds ~ 4*p^2. I've created from 1098 candidate p < 10000 a list of composite divisors. Only 260 of these Fib(p) are fully factored but the large unfactored composites in the remainder will not affect the result. There are 42 composite divisors of Fib(p) less than 4*10^8. These agree with the first 42 numbers in David's list. To validate all of his 148 terms (<10^10) would require checking additional candidate p < 50000.
On Feb 11, 2014, at 10:57 PM, David Wilson <davidwwilson at comcast.net> wrote:
> So, with the exception of p = 5, all prime divisors of Fib(p) are of the form kp +- 1.
> For example, Fib(19) has prime divisors 37 = 2*19-1 and 113 = 4*19-1.
> This implies that Fib(p) cannot be divisible by a prime smaller than 2p-1.
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