Moscow Mathematical Journal
Volume 17, Issue 2, April–June 2017 pp. 165–174.
On the extension of D(−8k2)-pair {8k2, 8k2+1}
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.
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:
Keywords: Diophantine m-tuples, Pell equations, elementary proofs.
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