## On Fibonacci numbers of the form $$q^ky^k$$. (Sur les nombres de Fibonacci de la forme $$q^ky^p$$.)(French)Zbl 1113.11010

The authors study the equations (1) $$F_n=q^ky^p$$ and (2) $$L_n=q^ky^p$$, where $$k>0$$ and $$p$$, $$q$$ are primes, and $$F_n$$ and $$L_n$$ denote the $$n$$th Fibonacci and Lucas numbers, respectively. They prove that: (i) if $$q=2$$, then the only solutions of (1) occur for $$n=3, 6, 12$$; (ii) If $$q$$ is odd and $$n$$ is even, then similarly $$n=0,4,12$$; (iii) if $$q$$ and $$n$$ are odd, then necessarily $$q\equiv 1\pmod 4$$. Furthermore, either $$q=n=5$$, or the Fibonacci entry point of $$q$$ must be odd, and (1) has a solution such that $$q\nmid n$$. The authors also show that if $$q=2$$, then (2) has a solution only when $$n=3$$.
The authors use the software package MAGMA to get solutions of the Diophantine equation arising in the solution of (2) for $$q=2$$ and $$y=3$$.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations

### Keywords:

Ribenboim’s question

Magma; PARI/GP
Full Text:

### References:

 [1] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, User’s guide to PARI-GP, version 2.1.1. Voir aussi http://www.parigp-home.de/; C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, User’s guide to PARI-GP, version 2.1.1. Voir aussi http://www.parigp-home.de/ [2] Bosma, W.; Cannon, J.; Playoust, C., The Magma System I: The User Language, J. Symb. Comp., 24, 235-265 (1997), Voir aussi · Zbl 0898.68039 [3] Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations, I. Fibonacci and Lucas Perfect Powers, soumis; Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations, I. Fibonacci and Lucas Perfect Powers, soumis · Zbl 1113.11021 [4] Cohn, J. H.E., Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc., 7, 24-28 (1965) · Zbl 0127.01902 [5] Ivorra, W., Sur les équations $$x^p + 2^\beta y^p = z^2$$ et $$x^p + 2^\beta y^p = 2 z^2$$, Acta Arith., 108, 327-338 (2003) · Zbl 1026.11035 [6] Ribenboim, P., The terms $$C x^h, (h \geqslant 3)$$ in Lucas sequences: an algorithm and applications to diophantine equations, Acta Arith., 106, 105-114 (2003) · Zbl 1162.11310
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