Adžaga, Nikola; Filipin, Alan On the extension of \(D(-8k^2)\)-pair \(\{8k^2, 8k^2 +1\}\). (English) Zbl 1433.11028 Mosc. Math. J. 17, No. 2, 165-174 (2017). Summary: Let \(k\) be a positive integer. The triple \(\{1, 8k^2, 8k^2 + 1\}\) has the property that the product of any two of its distinct elements subtracted by \(8k^2\) is a perfect square. By elementary means, we show that this triple can be extended to at most a quadruple retaining this property, i.e., if \(\{1, 8k^2, 8k^2 + 1, d\}\) has the same property, then \(d\) is uniquely determined \((d = 32k^2 + 1)\). Moreover, we show that even the pair \(\{8k^2, 8k^2 + 1\}\) can be extended in the same manner to at most a quadruple (the third and fourth element can only be \(1\) and \(32k^2 + 1)\). At the end, we suggest considering a similar problem of extending the triple \(\{1, 2k^2, 2k^2 + 2k + 1\}\) with a similar property as possible future research direction. Cited in 1 ReviewCited in 2 Documents MSC: 11D09 Quadratic and bilinear Diophantine equations 11A99 Elementary number theory Keywords:Diophantine \(m\)-tuples; Pell equations; elementary proofs PDFBibTeX XMLCite \textit{N. Adžaga} and \textit{A. Filipin}, Mosc. Math. J. 17, No. 2, 165--174 (2017; Zbl 1433.11028) Full Text: arXiv Link