# Moscow Mathematical Journal

Volume 17, Issue 2, April–June 2017 pp. 165–174.

On the extension of *D*(−8*k*^{2})-pair {8*k*^{2}, 8*k*^{2}+1}

**Authors**:
Nikola Adžaga (1) and Alan Filipin (1)

**Author institution:**(1) Department of Mathematics, Faculty of Civil Engineering, University of Zagreb, Kačićeva 26, Zagreb, Croatia

**Summary: **

Let *k* be a positive integer. The triple {1, 8*k*^{2}, 8*k*^{2} + 1}
has the property that the product of any two of its distinct elements
subtracted by 8*k*^{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, 8*k*^{2}, 8*k*^{2} + 1, *d*} has the same property, then *d* is
uniquely determined (*d* = 32*k*^{2} + 1). Moreover, we show that even the
pair {8*k*^{2}, 8*k*^{2} + 1} can be extended in the same manner to at most a
quadruple (the third and fourth element can only be 1 and 32*k*^{2} + 1).
At the end, we suggest considering a similar problem of extending the
triple {1, 2*k*^{2}, 2*k*^{2} + 2k + 1} with a similar property as possible future
research direction.

2010 Math. Subj. Class. 11D09, 11A99.

**Keywords:**Diophantine

*m*-tuples, Pell equations, elementary proofs.

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