%I #43 Dec 10 2016 19:33:56

%S 8,144

%N Fibonacci numbers that are the product of other Fibonacci numbers.

%C This sequence and A229037 and A235265 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - _T. D. Noe_, Jan 20 2014

%C Carmichael's theorem implies that 8 and 144 are the only terms of this sequence.

%C First two terms of A061899, A111687, A172150, A212703, and A231851. - _Omar E. Pol_, Jan 21 2014

%C Saha and Karthik conjectured (without reference to Carmichael's theorem) that the only positive integers k for which A001175(k^2) = A001175(k) are 6 and 12. (A000045(6) = 8 and A000045(12) = 144.) - _L. Edson Jeffery_, Feb 13 2014

%C Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only nontrivial perfect power Fibonacci numbers. - _Robert C. Lyons_, Dec 23, 2015

%H Yann Bugeaud, Maurice Mignotte, and Samir Siksek, <a href="http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n3-p05.pdf">Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers</a>, Annals of Mathematics, 163 (2006), pp. 969-1018.

%H Arpan Saha and Karthik C S, <a href="http://arxiv.org/abs/1102.1636">A few equivalences of Wall-Sun-Sun prime conjecture</a>, arXiv:1102.1636 [math.NT], 2011.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Carmichael%27s_theorem">Carmichael's theorem</a>.

%e The Fibonacci number 8 is in the sequence because 8=2*2*2, and 2 is a Fibonacci number that is not equal to 8. The Fibonacci number 144 is in the sequence because 144=3*3*2*2*2*2, and both 2 and 3 are Fibonacci numbers that are not equal to 144.

%Y Cf. A000045, A061899, A065108, A227875.

%K nonn,bref,fini,full,nice

%O 1,1

%A _Robert C. Lyons_, Jan 08 2014