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

Gupta, Hansraj; Singh, K.

Int. J. Math. Math. Sci. 8, 799-804 (1985).

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.

Reviewer: Betty Garrison (San Diego)

MSC:

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

Keywords:

congruence; Fibonacci sequence; perfect squares; k-triad sequence; recurrence relations