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


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


Zbl 0177.068
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