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

8, 144

Offset: 1

Author

Robert C. Lyons, Jan 08 2014

Keywords

• nonn
• bref
• fini
• full
• nice

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000045, A061899, A065108, A227875.