## Maximal frequencies of elements in second-order linear recurring sequences over a finite field.(English)Zbl 0747.11062

Let $$w(a,b)=(w)$$ be a second order linear recurring sequence over $$\mathbb{F}_ q$$ satisfying the relation $$w_{n+2}=aw_{n+1}-bw_ n$$ with initial terms $$w_ 0$$, $$w_ 1$$. Assuming that $$b\neq 0$$, $$w(a,b)$$ is purely periodic. If $$d\in\mathbb{F}_ q$$, let $$A(d)$$ denote the number of times that $$d$$ appears in a full period of the sequence $$w(a,b)$$. The main result of this paper is to prove that $$A(d)\leq\min(q,2\text{ord}(b))$$ for $$d\neq 0$$ and $$A(0)\leq\min(q- 1,2\text{ord}(b))$$. This result was proved in the special case of $$b=\pm 1$$, $$w_ 0=0$$, $$w_ 1=1$$ by the second and third author before.

### MSC:

 11T99 Finite fields and commutative rings (number-theoretic aspects) 11B37 Recurrences

### Keywords:

finite field; periods; linear recurring sequence
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