Primes having an incomplete system of residues for a class of second-order recurrences. (English) Zbl 0653.10005

Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San José/Cal., August 1986, 113-141 (1988).
(For the entire collection see Zbl 0635.00004.)
The sequence \(\{W_ n\}\) is defined by \(W_{n+2}=aW_{n+1}+bW_ n\), where \(a\) and the initial terms are integers, and \(b=\pm 1\). Those primes \(p\) for which all recurrences \(W_ n(a,\pm 1)\) are “defective”, i.e., they have an incomplete system of residues modulo \(p\), as \(a\) varies over all integers such that \(a^ 2+4b\not\equiv 0\pmod p\), are determined next. The following is shown.
1(a) If \(p\geq 5\) and \(a^ 2-4\not\equiv 0\pmod p\), then \(W_ n(a,-1)\) are all defective modulo \(p\).
(b) If \(p=2\) or \(3\), there exists a non-defective recurrence \(W_ n(a,-1)\) with \(a^ 2-4\not\equiv 0\pmod p\).
(c) If \(a^ 2-4\equiv 0\pmod p\) and \(W_ 0\equiv 0\), \(W_ 1\equiv 1\pmod p\), then \(W_ n(a,-1)\) is non-defective modulo \(p\).
2(a) If \(p>7\) and \(p\equiv 1\) or \(9\pmod {20}\) and \(a^ 2+4\not\equiv 0\pmod p\), then \(W_ n(a,1)\) are all defective modulo \(p\).
(b) For each \(p=2, 3, 5\) or \(7\), there exists \(a\) such that \(a^ 2+4\not\equiv 0\pmod p\) and \(W_ n(a,1)\) is non-defective modulo \(p\).
(c) If \(a^ 2+4\equiv 0\pmod p\) and \(W_ 0\equiv 0\), \(W_ 1\equiv 1\pmod p\), then \(W_ n(a,1)\) is non-defective modulo \(p\).
Reviewer: G. Philippou


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A07 Congruences; primitive roots; residue systems


Zbl 0635.00004