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A337231 Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.

%I A337231 #25 Nov 23 2023 12:02:22
%S A337231 231,323,377,1443,1551,1891,2737,2849,3289,3689,3827,4181,4879,5777,
%T A337231 6479,6601,6721,7743,8149,9879,10877,11663,13201,13981,15251,15301,
%U A337231 17119,17261,17711,18407,19043,20999,23407,25877,27071,27323,29281,30889,34561,34943,35207
%N A337231 Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.
%C A337231 If p is a prime, then A000045(p)^2==1 (mod p).
%C A337231 This sequence contains the odd composite integers for which the congruence holds.
%C A337231 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.
%C A337231 For a=1, b=-1, U(n) recovers A000045(n) (Fibonacci numbers).
%D A337231 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H A337231 Amiram Eldar, <a href="/A337231/b337231.txt">Table of n, a(n) for n = 1..1000</a>
%H A337231 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.3390/math9080838">On Generalized Lucas Pseudoprimality of Level k</a>, Mathematics (2021) Vol. 9, 838.
%H A337231 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).
%t A337231 Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] &]
%o A337231 (PARI) lista(nn) = my(list=List()); forcomposite(c=1, nn, if ((c%2) && (Mod(fibonacci(c), c)^2 == 1), listput(list, c))); Vec(list); \\ _Michel Marcus_, Sep 29 2023
%Y A337231 Cf. A000045.
%K A337231 nonn
%O A337231 1,1
%A A337231 _Ovidiu Bagdasar_, Aug 20 2020