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Moscow Mathematical Journal

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

On the extension of D(−8k2)-pair {8k2, 8k2+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


Let k be a positive integer. The triple {1, 8k2, 8k2 + 1} has the property that the product of any two of its distinct elements subtracted by 8k2 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, 8k2, 8k2 + 1, d} has the same property, then d is uniquely determined (d = 32k2 + 1). Moreover, we show that even the pair {8k2, 8k2 + 1} can be extended in the same manner to at most a quadruple (the third and fourth element can only be 1 and 32k2 + 1). At the end, we suggest considering a similar problem of extending the triple {1, 2k2, 2k2 + 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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