Fibonacci Prime
There are an infinite number of primes among the Fibonacci sequence numbers Un.
Proof:
hcf (Up, n) = 1, for all n<p [by Lemma 1]
let p->99999… [Euclid]
=> Up prime
=> there exist an infinite number of prime Up
LEMMA 1
Any divisor of Up is greater than p [p odd prime <>5]
Proof:
q | U(q+/-1) [q odd prime <>5][by Lemma 2]
=> Up | U(q+/-1) [by Lemmas 3 & 4]
=> p | q+/-1
=> p<q [since p,q both odd]
LEMMA 2
q prime <>5 => q | U(q-1) or q | U(q+1)
LEMMA 3
q is a primitive divisor of Up
Proof:
Suppose q | Up, with n<p [p prime]
hcf (Un,Up) = U(hcf (n,p)) = U1 = 1
=> Un<>0 mod q, i.e. q primitive divisor of Up
LEMMA 4
If q is a primitive divisor of Un and q | Um, then Un | Um
Proof:
If q is a primitive divisor of Un, then q | Unj only
n | nj => Un | Unj