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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Sets in which $ xy+k$ is always a square

Author: Ezra Brown
Journal: Math. Comp. 45 (1985), 613-620
MSC: Primary 11D57
MathSciNet review: 804949
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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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PII: S 0025-5718(1985)0804949-7
Article copyright: © Copyright 1985 American Mathematical Society

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