## Special Lucas sequences, including the Fibonacci sequence, modulo a prime.(English)Zbl 0716.11009

A tribute to Paul Erdős, 349-357 (1990).
For a fixed integer $$a$$ let $$u_n$$ be a Lucas sequence defined by $$u_n=au_{n-1}+u_{n-2}$$ with $$u_0=0$$ and $$u_1=1$$ and for a fixed prime $$p$$ let $$\bar u_n$$ be the sequence defined by the same way in $$\mathbb F_ p$$, the finite field with $$p$$ elements. Let $$k(p)$$ be the length of the shortest period of the sequence $$\bar u_n$$. The author proves the following theorem:
Let $$S=S(p)$$ be the set of frequencies with which different residues occur in the sequence $$\bar u_n$$ $$(0\le n<k(p))$$. For $$p>7$$ and $$p \nmid a(a^2+4)$$ we have $$S=\{0,1,2\}$$ or $$\{0,1,2,3\}$$ if $$k(p)\not\equiv 0\pmod 4$$, $$=\{0,2,4\}$$ if $$k(p)\equiv 4\pmod 8$$; $$=\{0,1,2\}$$ or $$\{0,2,3\}$$ or $$\{0,1,2,4\}$$ or $$\{ ,2,3,4\}$$ if $$k(p)\equiv 0\pmod 8$$.
From this result some consequences also follow concerning the distribution properties of the sequences $$u_n$$ modulo $$p$$.
[For the entire collection see Zbl 0706.00007.]
Reviewer: Péter Kiss (Eger)

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B50 Sequences (mod $$m$$) 11T99 Finite fields and commutative rings (number-theoretic aspects)

Zbl 0706.00007