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On a conjecture concerning sequences of the third order. (English) Zbl 0583.10007
Let $$\{a_ n\}$$ be the sequence of integers defined by the relation $$a_{n+3}=a_{n+1}+a_ n$$ with initial conditions $$a_ 0=3$$, $$a_ 1=0$$, $$a_ 2=2$$. It is easy to see that if n is a prime, then n $$| a_ n$$. The authors show that the converse is not true, i.e., there are composite numbers n such that n $$| a_ n$$ (e.g. $$n=3037 \cdot 9109).$$ They got the counter-examples by the following theorem: Let p,q be primes such that (i) $$q=1+k(p-1)$$, $$k>1$$; (ii) p,q are representable by the quadratic form $$x^ 2+23y^ 2$$; (iii) all roots of the polynomial $$x^ 3-x-1$$ in $${\mathbb{Z}}/q{\mathbb{Z}}$$ are k-th power residues modulo q. Then $$a_{pq}\equiv 0$$ (mod pq).
Reviewer: P.Kiss

##### MSC:
 11B37 Recurrences
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##### References:
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