- The folling is a conjecture, I've come up with.

Prime Fibonacci and Lucas numbers, and their prime factors,2kp+/-1.

When p= a small prime

F(p) is prime, when it has only one(itelf) prime factor, in the form

2kp+/-1.

Composite F(p) contain at least one prime factor in the form 2kp+/-1.

L(p) is prime, when it has only one(itself) prime factor, in the form

2kp+1.

Composite L(p) contain at least one prime factor in the form 2kp+1.

To decide +/-1, for the prime fibonacci's consider p, ? mod 10.

If p ends in 3, or 7, F(p)=2kp -1

If p ends in 1, or 9, F(p)=2kp +1

Quadratic residue ?

After F(5)=5, which ofcourse can't leave a remainder so, there is no

+/-1 given.

Prime string .010001000101101111100101101111...

I'd like to thank, David Broadhurst for pointing out the mod 10

connection.

Take into consideration that:

F(p)= Phi^p + phi^p / sqrt(5)

L(p)= Phi^p - phi^p

This emulates Mersenne numbers in the form 2^p -1.

Mersenne primes have only one(itself) prime factor, in the form 2kp+1.

Composite M(p) must contain prime factors in the form 2kp+1.

It may or may not be trivial that:

When F(p) is prime, p is also prime.(exception F(2^2), a consequence

of F(1)=F(2) = 1.)

When L(p) is prime p is also prime.(exceptions L(2^1)L(2^2)L(2^3)and,

L(2^4) are powers of 2 !)

These, L(2^k) are in the form 2kp-1.

Known Fibonacci primes for small p.

Notice the relation to prime string .010001000101101111100101101111...

p, prime string

3

5 .

7 0

11 1

13 0

17 0

23 0

29 1

43 0

47 0

83 0

131 1

137 0

359 1

431 1

433 0

449 1

509 1

569 1

571 1

2971 1

4723 0

5387 0

9311 1

9677 0

14431 1

25561 1

30757 0

35999 1

37511 1

81839 1

104911 1 Verified on 4 / 2001

I am open for argument, questions, corrections !

My hope is to find a pattern in prime string, or come up with an easy

primality testing method like that of Mersenne primes.

Shane F. - Shane Findley noted the following p for which

F(p) is a titanic PrP:

> 5387 0

The PrP F(104911) was discovered by Bouk de Water. See http://

> 9311 1

> 9677 0

> 14431 1

> 25561 1

> 30757 0

> 35999 1

> 37511 1

> 81839 1

> 104911 1 Verified on 4 / 2001

ourworld.compuserve.com/homepages/hlifchitz/Henri/fr-us/PrpRec.htm

A proof of primality of F(104911) is

far beyond the present ability of humankind.

These titanic Fibs are proven prime:

F(81839) 17103 p54 2001

F(35999) 7523 p54 2001

F(30757) 6428 p54 2001

F(25561) 5342 p54 2001

F(14431) 3016 p54 2001

F(9677) 2023 c2 2000

F(9311) 1946 DK 1995

F(5387) 1126 WM 1990

Shane noted that F(37511) is PrP but he missed F(50833).

Bouk and I have been worked intensively on these two.

Their proofs will be very difficult to complete,

but we live in hope of proving F(37511), for which

the present factorization percentages of the digits

of N-1 and N+1 are 22.31% and 7.92%

David Broadhurst