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A337232 Even composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.

%I A337232 #15 Nov 23 2023 12:02:07
%S A337232 4,8,14,22,26,34,38,46,58,62,74,82,86,88,94,106,118,122,134,142,146,
%T A337232 158,166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,
%U A337232 334,346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514,526
%N A337232 Even composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.
%C A337232 If p is a prime, then A000045(p)^2 == 1 (mod p).
%C A337232 This sequence contains the even integers for which the congruence holds.
%C A337232 The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2) = a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p) == 1 (mod p) whenever p is prime and b=1,-1.
%C A337232 For a=1, b=-1, U(n) recovers A000045(n) (Fibonacci numbers).
%C A337232 No terms divisible by 3. - _Robert Israel_, Sep 15 2020
%D A337232 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H A337232 Robert Israel, <a href="/A337232/b337232.txt">Table of n, a(n) for n = 1..10000</a>
%H A337232 D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).
%p A337232 select(t -> combinat:-fibonacci(t) &^ 2 - 1 mod t = 0, 2*[$2..1000]); # _Robert Israel_, Sep 15 2020
%t A337232 Select[Range[2, 2000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] &]
%Y A337232 Cf. A000045, A337231 (odd terms).
%K A337232 nonn
%O A337232 1,1
%A A337232 _Ovidiu Bagdasar_, Aug 20 2020