## 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

### MSC:

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

Zbl 0635.00004