Sets in which $$xy+k$$ is always a square.(English)Zbl 0577.10015

A $$P_ k$$-set of size n is a set $$\{x_ 1,...,x_ n\}$$ of distinct positive integers such that $$x_ ix_ j+k$$ is a perfect square whenever $$i\neq j$$. A $$P_ k$$-set X can be extended if there exists $$y\not\in X$$ such that $$X\cup \{y\}$$ is still a $$P_ k$$-set. The most famous result on $$P_ k$$-sets is due to A. Baker and H. Davenport [Q. J. Math., Oxf. II. Ser. 20, 129-137 (1969; Zbl 0177.068)], who proved that the $$P_ 1$$-set $$\{$$ 1,3,8,120$$\}$$ cannot be extended.
In this paper it is shown that if $$k\equiv 2 (mod 4)$$, then there does not exist a $$P_ k$$-set of size 4 and that the $$P_{-1}$$-set $$\{$$ 1,2,5$$\}$$ cannot be extended. Some other specialized results of $$P_ k$$- sets are also proven.
Reviewer: E.L.Cohen

MSC:

 11D09 Quadratic and bilinear Diophantine equations 11B37 Recurrences 11A07 Congruences; primitive roots; residue systems

Zbl 0177.068
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