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The Fibonacci-norm of a positive integer: observations and conjectures. (English) Zbl 1210.11024

The authors introduce the notion of the Fibonacci norm of a positive integer $n$ as being the smallest positive integer $k$ such that $n$ divides the $k$th Fibonacci number ${F}_{k}$, state a few conjectures concerning this function and prove a few results. However, this function already exists in the literature under several names, most commonly known as the order (or rank) of appearance (or the order of apparition, or the Fibonacci entry point) and it has been extensively studied by E. Lucas, R. D. Carmichael, D. H. Lehmer, as well as more modern researchers.

In particular, Conjecture 2.1 in the paper under review is a well-known theorem, and so is Conjecture 6.4. Both these results can be found in the classical [E. Lucas, Théorie des fonctions numériques simplement périodiques, Am. J. Math. 1, 184–240, 289–321 (1878; JFM 10.0134.05) (for the proof of Conjecture 2.1, see the Théorème Fondamental of Section XXVI, page 300, while for the proof of Conjecture 6.4 see the Théorème of Section XIII, page 210).

Conjectures 6.5 and 6.6 are known to be false (the involved ratios there do not tend to infinity, in fact they are always 1, 2 or 4; see, for example, Theorem 1 in H. Wilcox, Fibonacci Q. 24, 356–361 (1986; Zbl 0603.10011)), and the truth of Conjecture 6.7 follows from sieve methods and a well-known result of Lagarias which says that the set of primes having odd order of appearance is of density $1/3$ [J. C. Lagarias, Pac. J. Math. 118, 449–461 (1985; Zbl 0569.10003); errata, Pac. J. Math. 162, 393–396 (1994; Zbl 0790.11014)].

The paper also contains a few results and conjectures about the local behavior of the index of appearance.

MSC:
 11B39 Fibonacci and Lucas numbers, etc.