A235383 Fibonacci numbers that are the product of other Fibonacci numbers.

8, 144

Offset

1,1

Comments

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

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

First two terms of A061899, A111687, A172150, A212703, and A231851. - Omar E. Pol, Jan 21 2014

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

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

Links

Table of n, a(n) for n=1..2.

Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 969-1018.

Arpan Saha and Karthik C S, A few equivalences of Wall-Sun-Sun prime conjecture, arXiv:1102.1636 [math.NT], 2011.

Wikipedia, Carmichael's theorem.

Example

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.

Crossrefs

Cf. A000045, A061899, A065108, A227875.

Sequence in context: A320396 A112464 A227875 * A275034 A275139 A370071

Adjacent sequences: A235380 A235381 A235382 * A235384 A235385 A235386

Keyword

nonn,bref,fini,full,nice

Author

Robert C. Lyons, Jan 08 2014

Status

approved