Naslov
On the extension of D(-8k^2)-triple {;;;1, 8k^2, 8k^2+1};;;

Autori
Adžaga, Nikola

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Conference on Elementary and analytic number theory (ELAZ 2016) / Elsholtz, Christian ; Nowak, Georg ; Tichy, Robert - , 2016, 9-9

Skup
Elementare und Analytische Zahlentheorie, Conference on elementary and analytic number theory (ELAZ 2016)

Mjesto i datum
Strobl, Austrija, 05.09.2016. - 09.09.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
Diophantine m-tuples ; D(n)-m-tuples

Sažetak
By elementary means, we show that the D(-8k^2)-triple {; ; ; 1, 8k^2, 8k^2+1}; ; ; can be extended to at most a quadruple (the fourth element can be only 32k^2+1). A set of m positive integers {; ; ; a_1, a_2, ..., a_m}; ; ; is called D(n)-m-tuple if a_i a_j+n is a perfect square for all 1 <= i < j <= m. Extending the initial triple with d and then eliminating d leads to a system consisting of a Pell (z^2-(16k^2+2)y^2=1) and a pellian equation (x^2-2y^2=-8k^2+1). By solving Pell equation, we get two recurrent sequences y_n and z_n. Due to the second equation, the problem reduces to examining when can an element of the new sequence X_n = 2y_n^2-8k^2+1 be a complete square. Using the relations between y_n and z_n, e.g. y_{; ; ; 2n+1}; ; ; = 2y_nz_n, we write X_n as a product of two factors, one of which is obviously not a square. We finish the proof by showing that these factors are relatively prime via principle of descent.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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