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Generalization of a problem of Diophantus. (English) Zbl 0849.11018

Let \(n\) be an integer. The set of natural numbers \(\{a_1, a_2, \dots, a_m\}\) has the property of Diophantus of order \(n\), in brief \(D(n)\), if \(i\neq j\) for all \(i,j= 1, 2,\dots, m\) and the following holds: \(a_i a_j+ n= b^2_{ij}\), where \(b_{ij}\) is an integer. Some problems of the existence of sets of four natural numbers with property \(D(n)\), for any integer \(n\), are considered. The main results are: (1) If \(n\) is an integer of the form \(n= 4k+2\) (\(k\) an integer), then there does not exist a set of four natural numbers with the property \(D(n)\); (2) If \(n\) is an integer not of the form \(4k+2\), and \(n\not\in S=\{ 3, 5, 8, 12, 20, -1, -3, -4\}\), then there exists at least one set of four natural numbers with the property \(D(n)\).
Reviewer: E.L.Cohen (Ottawa)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D09 Quadratic and bilinear Diophantine equations
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