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On the extensibility of \(D(4)\)-pair \(\{ k-2, k+2\}^*\). (English) Zbl 1360.11057

Summary: In this paper we prove that if \(k\geq 3\), \(c\), and \(d\) are positive integers with \(c<d\) and the set \(\{k-2, k+2, c, d\}\) has the property that the product of any of its distinct elements increased by 4 is a perfect square, then \(d\) is uniquely determined.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11D79 Congruences in many variables
11J86 Linear forms in logarithms; Baker’s method

Keywords:

extensibility
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