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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:

[1] БОРЕВИЧ З. И.-ШАФАРЕВИЧ И. Р.: Теория чисел. Знание. Москва 1972. · Zbl 1156.34335 · doi:10.1070/PU1972v014n05ABEH004669
[2] ДЕЛОНЕ Б. Н.-ФАДДЕЕВ Д. К.: Теория иррациональностей третьей степени. Труды матем. института имени В. А. Стеклоа 11. Москва 1940. · Zbl 0158.02905
[3] DICKSON L. E.: Modern Elementary Theory of Numbers. The University of Chicago Press. Chicago 1939. Second Impression 1943. · Zbl 0027.29502
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