## 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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