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

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 $$n\geq 1$$.

### 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