A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611

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". Crandall and Pomerance state that the first such Frobenius pseudoprime is actually 4181.

The Frobenius (1,-1) pseudoprimes are a subset of the odd Fibonacci pseudoprimes A081264. Among other ways, this can be seen by Theorem 3.6.6 of Crandall and Pomerance (2005) where the Frobenius criterion with respect to x^2 - Px + Q is an additional condition on an input which has passed the Lucas test for the same polynomial. - Dana Jacobsen, Aug 05 2015

Many other quadratics have a sparser set of pseudoprimes. For example, while there are 98702 pseudoprimes below 10^13 with respect to the Fibonacci polynomial, there are only 3897 for x^2 - 3x - 5. - Dana Jacobsen, Aug 05 2015

This is the intersection of A049062 and (A081264 union A141137), that is, composite k coprime to 5 such that Fibonacci(k) == (k/5) (mod k) and that k divides Fibonacci(k-(k/5)), where (k/5) is the Legendre or Jacobi symbol. - 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..10000 (from Dana Jacobsen's site, terms 1..653 from Max Alekseyev)
• Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 89.
• Jon Grantham, Frobenius pseudoprimes, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
• Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms through 10^13)
• A. Rotkiewicz, Lucas and Frobenius Pseudoprimes, Annales Mathematicae Silesiane, 17 (2003): 17-39.
• Lawrence Somer, Lucas sequences {Uk} for which U2p and Up are pseudoprimes for almost all primes p, Fibonacci Quart. 44 (2006), no. 1, 7-12.
• Eric W. Weisstein, Frobenius Pseudoprime, MathWorld.

Crossrefs

Cf. A049062, A081264, A141137.

Cf. also A005845, A094063, A094395, A094411.

Terms congruent to 2 or 3 mod 5 are given in A212423.

Programs

PARI

{ isFP(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)||(kronecker(5,n)==1 && t==x) }

Perl

use ntheory ":all"; foroddcomposites { say if is_frobenius_pseudoprime($,1,-1) } 1e10; # _Dana Jacobsen, Aug 05 2015