Fibonacci primitive roots and Wall’s question. (English) Zbl 0936.11011

Li, Hua-Chieh

Fibonacci Q. 37, No. 1, 77-84 (1999).

Define \(g\) to be a Fibonacci primitive root \(\pmod m\) if \(g\) is a primitive root \(\pmod m\) such that \(g^2-g-1\equiv 0\pmod m\). Let \(k(m)\) be the period of the Fibonacci sequence \(\pmod m\), that is, the least positive integer such that \(F_{k(m)}\equiv 0\pmod m\), \(F_{k(m)+1}\equiv 1\pmod m\). The author’s main result states that if \(p\) is a prime such that \(k(p)\neq k(p^2)\), then the existence of a Fibonacci primitive root \(\pmod p\) implies the existence of a Fibonacci primitive root \(\pmod {p^n}\) for all \(n\geq 1\).

Reviewer: N.Robbins (San Francisco)

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11A07	Congruences; primitive roots; residue systems
11B50	Sequences (mod \(m\))

Keywords:

Fibonacci primitive root; Fibonacci sequence; period