A227875 Fibonacci numbers which are perfect powers.

0, 1, 8, 144

Offset: 1

Author

Jean-François Alcover, Oct 25 2013

Keywords

• nonn
• bref
• fini
• full

Comments

Also, Fibonacci numbers which are products of Fibonacci numbers (each greater than 1 when the product is greater than 1 - see A235383). - Rick L. Shepherd, Feb 19 2014

The terms of the subsequence (1, 8, 144) are the Fibonacci numbers that are powerful numbers. - Robert C. Lyons, Jul 12 2016

Also Fibonacci numbers without any primitive divisors. See [Heuberger & Wagner]. - Michel Marcus, Aug 21 2016

It was proved (Bugeaud, Mignotte, and Siksek, 2006, p. 971) that the only perfect powers among the Fibonacci numbers and Lucas numbers are {0, 1, 8, 144} and {1, 4}, respectively. - Daniel Forgues, Apr 09 2018

Links

• Vladica Andrejic, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 38-44.
• Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
• 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.
• Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016. Gives the complement sequence w.r.t Fibonacci numbers.
• J. Mc Laughlin, Small prime powers in the Fibonacci sequence, arXiv:math/0110150 [math.NT] (2001).
• Attila Pethő, Diophantine properties of linear recursive sequences II, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 17:2 (2001), pp. 81-96.

Crossrefs

Cf. A000045, A001597, A072381, A114842.

Programs

Mathematica

perfectPowerQ[0] = True; perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; Union[Select[Fibonacci /@ Range[0, 20], perfectPowerQ]]