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On \(k\)-triad sequences. (English) Zbl 0585.10006

For any integer \(k\) the integers \(a_ 1\), \(a_ 2\), \(a_ 3\) are a \(k\)-triad if \(a_ 1a_ 2+k\), \(a_ 1a_ 3+k\), \(a_ 2a_ 3+k\) are all perfect squares. An ascending sequence of integers \(a_ 1,a_ 2,a_ 3,\ldots,a_ n,\ldots\) is a \(k\)-triad sequence if every three consecutive elements of the sequence form a \(k\)-triad. Whenever \(a_ 1<a_ 2\), \(a_ 1a_ 2+k=c_ 1^ 2\), it is shown that a \(k\)-triad sequence \(a_ 1,a_ 2,a_ 3,\ldots,a_ n,\ldots\) exists and recurrence relations are given so that \(a_ n\) and the associated \(c_{n-1}\) such that \(a_{n-1}a_ n+k=c_{n- 1}^ 2\) may be easily calculated. It is also proved that if \(a_ 1,a_ 2,a_ 3\) is a \(k\)-triad and \(k\equiv 2\pmod 4\), then there is no integer \(a\) for which \(a_ 1a+k\), \(a_ 2a+k\), \(a_ 3a+k\) are all perfect squares.

MSC:

11B37 Recurrences
11A07 Congruences; primitive roots; residue systems
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