FIBONACCI PSEUDOPRIME

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.

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References

External links

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.

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.

★ MathWorld: Fibonacci Pseudoprime