We show that if m > 1 is a Fibonacci number such that ϕ(m) | m-1, where ϕ is the Euler function, then m is prime
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-1-2, author = {Florian Luca}, title = {Fibonacci Numbers with the Lehmer Property}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {55}, year = {2007}, pages = {7-15}, zbl = {1112.11007}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-1-2} }
Florian Luca. Fibonacci Numbers with the Lehmer Property. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 7-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-1-2/