Fibonacci pseudoprime
In number theory, a pseudoprime is a number that passes some test that all primess 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
- Vn is congruent to P mod n.
Here the notation refers to the Lucas sequence with parameters P, Q producing a series of numbers Un, Vn. It is conjectured that there are no even Fibonacci pseudoprimes (see Somer). A strong Fibonacci pseudoprime may defined as follows (see Müller and Oswald): - An odd composite integer n is also a Carmichael number
- 2(pi + 1) | (n − 1) or 2(pi + 1) | (n − pi) for every prime pi dividing n.
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, Larence. "On Even Fibonacci Pseudoprimes." In G.E. Bergum et al, eds. Applications of Fibonacci Numbers. Volume 4. Dordrecht: Kluwer, 1991. 277-288.
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.
- Dénes, Tamás. On the connections between RSA cryptosystem and the Fibonacci numbers.
- MathWorld: Fibonacci Pseudoprime
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