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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 \(m\))
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