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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)

Citations:

Zbl 0706.00007