A212423 Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

5777, 10877, 75077, 100127, 113573, 161027, 162133, 231703, 430127, 635627, 851927, 1033997, 1106327, 1256293, 1388903, 1697183, 2263127, 2435423, 2662277, 3175883, 3399527, 3452147, 3774377, 3900797, 4109363, 4226777, 4403027, 4828277, 4870847

Offset: 1

Author

Max Alekseyev, May 16 2012

Keywords

• nonn

Comments

Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2 - x - 1 is 5777". However n = 5777 is the first Frobenius pseudoprime with respect to x^2 - x - 1 that has Jacobi symbol (5/n) = -1, i.e., n == 2,3 (mod 5). Unrestricted version with the first term 4181 is given in A212424.

Intersection of A212424 and A047221.

Composite k == 2,3 (mod 5) such that Fibonacci(k) == -1 (mod k) and that k divides Fibonacci(k+1). - Jianing Song, Sep 12 2018

References

• R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..14070 (terms below 10^13 from Dana Jacobsen's site)
• Jon Grantham, Frobenius Pseudoprimes, Mathematics of Computation, 7 (2000), 873-891.
• Dana Jacobsen, Pseudoprime Statistics, Tables, and Data.
• Eric Weisstein's World of Mathematics, Frobenius Pseudoprime.

Crossrefs

Cf. A047221, A094063, A094395, A094411, A212424.

Programs

PARI

{ isFP23(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x) }