Sets in which 𝑥𝑦+𝑘 is always a square

Brown, Ezra

doi:10.1090/S0025-5718-1985-0804949-7

Sets in which $xy+k$ is always a square
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Math. Comp. 45 (1985), 613-620 Request permission

Abstract:

A ${P_k}$-set of size n is a set $\{ {x_1}, \ldots ,{x_n}\}$ of distinct positive integers such that ${x_i}{x_j} + k$ is a perfect square, whenever $i \ne j$; a ${P_k}$-set X can be extended if there exists $y \notin X$ such that $X \cup \{ y\}$ is still a ${P_k}$-set. The most famous result on ${P_k}$-sets is due to Baker and Davenport, who proved that the ${P_1}$-set 1, 3, 8, 120 cannot be extended. In this paper, we show, among other things, that if $k \equiv 2\;\pmod 4$, then there does not exist a ${P_k}$-set of size 4, and that the ${P_{ - 1}}$-set 1, 2, 5 cannot be extended.

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