Fibonacci pseudoprime

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In number theory, a pseudoprime is a number that passes some test that all primes pass, but is actually composite. A Fibonacci pseudoprime is a composite integer n that satisfies the following conditions:

P > 0 and Q = +1 or −1
V_n is congruent to P mod n.

Here the notation refers to the Lucas sequence with parameters P, Q producing a sequence of numbers U_n, V_n.

It is conjectured that there are no even Fibonacci pseudoprimes (see Somer).

A strong Fibonacci pseudoprime may be defined as follows (see Müller and Oswald):

An odd composite integer n is also a Carmichael number
2(p_i + 1) | (n − 1) or 2(p_i + 1) | (n − p_i) for every prime p_i dividing n.

The smallest example of a strong Fibonacci pseudoprime is 443372888629441, which has factors 17, 31, 41, 43, 89, 97, 167 and 331.

[edit] References

Müller, Winfired B. and Alan Oswald. "Generalized Fibonacci Pseudoprimes and Probable Primes." In G.E. Bergum et al, eds. Applications of Fibonacci Numbers. Volume 5. Dordrecht: Kluwer, 1993. 459-464.
Somer, Lawrence. "On Even Fibonacci Pseudoprimes." In G.E. Bergum et al, eds. Applications of Fibonacci Numbers. Volume 4. Dordrecht: Kluwer, 1991. 277-288.