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

%I A212423 #33 Feb 16 2025 08:33:17
%S A212423 5777,10877,75077,100127,113573,161027,162133,231703,430127,635627,
%T A212423 851927,1033997,1106327,1256293,1388903,1697183,2263127,2435423,
%U A212423 2662277,3175883,3399527,3452147,3774377,3900797,4109363,4226777,4403027,4828277,4870847
%N A212423 Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.
%C A212423 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.
%C A212423 Intersection of A212424 and A047221.
%C A212423 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
%D A212423 R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.
%H A212423 Amiram Eldar, <a href="/A212423/b212423.txt">Table of n, a(n) for n = 1..14070</a> (terms below 10^13 from Dana Jacobsen's site)
%H A212423 Jon Grantham, <a href="https://doi.org/10.1090/S0025-5718-00-01197-2">Frobenius Pseudoprimes</a>, Mathematics of Computation, 7 (2000), 873-891.
%H A212423 Dana Jacobsen, <a href="http://ntheory.org/pseudoprimes.html">Pseudoprime Statistics, Tables, and Data</a>.
%H A212423 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FrobeniusPseudoprime.html">Frobenius Pseudoprime</a>.
%o A212423 (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) }
%Y A212423 Cf. A047221, A094063, A094395, A094411, A212424.
%K A212423 nonn
%O A212423 1,1
%A A212423 _Max Alekseyev_, May 16 2012