engleski

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

Let k be a positive integer. 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.

Diophantine m-tuples, Pell equations, elementary proofs

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