for which congruence () holds for Q = −1 and all P. It follows that an odd composite integer n is a strong Fibonacci pseudoprime if and only if: n is a Carmichael number 2(p + 1) | (n − 1) or 2(p + 1) | (n − p) for every prime p dividing n. The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331. Pell pseudoprimes A Pell pseudoprime''' may be defined as a composite number n for which equation () above is true with P = 2 and Q = −1; the sequence Un then being the Pell sequence. The first pseudoprimes are then 35, 169, 385, 779, 899, 961, 1121, 1189, 2419, ... This differs from the definition in which may be written as: with (P, Q) = (2, −1) again defining Un as the Pell sequence. The first pseudoprimes are then 169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215 ... A third definition uses equation (5) with (P, Q'') = (2, −1), leading to the pseudoprimes 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ... References External links Anderson, Peter G. Fibonacci Pseudoprimes, their factors, and their entry points. Anderson, Peter G. Fibonacci Pseudoprimes under 2,217,967,487 and their factors. Jacobsen, Dana Pseudoprime Statistics, Tables, and Data (data for Lucas, Strong Lucas, AES Lucas, ES Lucas pseudoprimes below 1014; Fibonacci and Pell pseudoprimes below 1012) Fibonacci numbers Pseudoprimes