Factorization Theorems

Here we collect theorems about the primitive factors of the Fibonacci and Lucas numbers.

Theorem ([D-2000])
Let p > 7 be a prime satisfying the following two conditions:

1. p = 2 (mod 5) or p = 4 (mod 5)
2. 2p-1 is also prime.
Then Fp is composite, in fact (2p-1) | Fp.

Theorem ([J-1973, p. 11])
Let n be odd and let p be an odd, primitive prime divisor of Fn. Then

1. p = 1 (mod 4).
2. if p = 1 or -1 (mod 10) then p = 1 (mod 4n).
3. if p = 3 or -3 (mod 10) then p = 2n-1 (mod 4n).

Theorem ([J-1973, p. 11])
Let n be positive and let p be an odd, primitive prime divisor of Ln. Then

1. if p = 1 or -1 (mod 10) then p = 1 (mod 2n).
2. if p = 3 or -3 (mod 10) then p = -1 (mod 2n).

If anyone knows of other theorems, please let me know.


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