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Pregled bibliografske jedinice broj: 833826

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


Nikola Adžaga
On the extension of D(-8k^2)-triple {;1, 8k^2, 8k^2+1}; // Conference on Elementary and analytic number theory (ELAZ 2016) / Christian Elsholtz, Georg Nowak, Robert Tichy (ur.).
Strobl, Austrija, 2016. str. 9-9 (predavanje, nije recenziran, sažetak, znanstveni)


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

Autori
Nikola Adžaga

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

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

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

Mjesto i datum
Strobl, Austrija, 05-09.09.2016

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



POVEZANOST RADA


Projekti:
HRZZ-IP-2013-11-6422 - Diofantove m-torke, eliptičke krivulje, Thueove i indeksne jednadžbe (DIOPHANTINE) (Dujella, Andrej, HRZZ - 2013-11) ( POIROT)

Ustanove:
Građevinski fakultet, Zagreb,
Prirodoslovno-matematički fakultet, Zagreb

Profili:

Avatar Url Nikola Adžaga (autor)

Citiraj ovu publikaciju

Nikola Adžaga
On the extension of D(-8k^2)-triple {;1, 8k^2, 8k^2+1}; // Conference on Elementary and analytic number theory (ELAZ 2016) / Christian Elsholtz, Georg Nowak, Robert Tichy (ur.).
Strobl, Austrija, 2016. str. 9-9 (predavanje, nije recenziran, sažetak, znanstveni)
Nikola Adžaga (2016) On the extension of D(-8k^2)-triple {;1, 8k^2, 8k^2+1};. U: Christian Elsholtz, Georg Nowak, Robert Tichy (ur.)Conference on Elementary and analytic number theory (ELAZ 2016).
@article{article, year = {2016}, pages = {9-9}, keywords = {Diophantine m-tuples, D(n)-m-tuples}, title = {On the extension of D(-8k\^{}2)-triple {;1, 8k\^{}2, 8k\^{}2+1};}, keyword = {Diophantine m-tuples, D(n)-m-tuples}, publisherplace = {Strobl, Austrija} }
@article{article, year = {2016}, pages = {9-9}, keywords = {Diophantine m-tuples, D(n)-m-tuples}, title = {On the extension of D(-8k\^{}2)-triple {;1, 8k\^{}2, 8k\^{}2+1};}, keyword = {Diophantine m-tuples, D(n)-m-tuples}, publisherplace = {Strobl, Austrija} }




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