On Fibonacci numbers with few prime divisors
Bugeaud, Yann ; Luca, Florian ; Mignotte, Maurice ; Siksek, Samir
Proc. Japan Acad. Ser. A Math. Sci., Tome 81 (2005) no. 3, p. 17-20 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci number, and $\omega(n)$ for the number of distinct prime divisors of $n$. We give a description of Fibonacci numbers satisfying $\omega(F_n) \leq 2$. Moreover, we prove that the inequality $\omega(F_n) \geq (\log n)^{\log 2 + o(1)}$ holds for almost all $n$. We conjecture that $\omega(F_n) \gg \log n$ for composite $n$, and give a heuristic argument in support of this conjecture.
Publié le : 2005-02-14
Classification:  Fibonacci numbers,  arithmetic functions,  prime divisors,  11B39,  11K65 
@article{1116442053,
     author = {Bugeaud, Yann and Luca, Florian and Mignotte, Maurice and Siksek, Samir},
     title = {On Fibonacci numbers with few prime divisors},
     journal = {Proc. Japan Acad. Ser. A Math. Sci.},
     volume = {81},
     number = {3},
     year = {2005},
     pages = { 17-20},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1116442053}
}

Bugeaud, Yann; Luca, Florian; Mignotte, Maurice; Siksek, Samir. On Fibonacci numbers with few prime divisors. Proc. Japan Acad. Ser. A Math. Sci., Tome 81 (2005) no. 3, pp.  17-20. http://gdmltest.u-ga.fr/item/1116442053/