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Scientific paper - Original scientific paper
On the extensions of the Diophantine triples in Gaussian integers
Monatshefte für Mathematik, 197(4) (2022); 535-563. urn:nbn:hr:237:781222
Adžaga, Nikola; Filipin, Alan; Franušić, Zrinka

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Adžaga, N., Filipin, A. & Franušić, Z. (2022). On the extensions of the Diophantine triples in Gaussian integers. Monatshefte für Mathematik, 197(4)., 535-563. Retrieved from https://urn.nsk.hr/urn:nbn:hr:237:781222

Adžaga, Nikola, et al. "On the extensions of the Diophantine triples in Gaussian integers." Monatshefte für Mathematik, vol. 197(4), 2022, pp. 535-563. https://urn.nsk.hr/urn:nbn:hr:237:781222

Adžaga, Nikola, Alan Filipin and Zrinka Franušić. "On the extensions of the Diophantine triples in Gaussian integers." Monatshefte für Mathematik 197(4) (2022): 535-563. https://urn.nsk.hr/urn:nbn:hr:237:781222

Adžaga, N., Filipin, A. and Franušić, Z. (2022) 'On the extensions of the Diophantine triples in Gaussian integers', Monatshefte für Mathematik, 197(4), pp. 535-563. Available at: https://urn.nsk.hr/urn:nbn:hr:237:781222 (Accessed 04 November 2024)

Adžaga N, Filipin A, Franušić Z. On the extensions of the Diophantine triples in Gaussian integers. Monatshefte für Mathematik [Internet]. 2022 [cited 2024 November 04];197(4):535-563. Available at: https://urn.nsk.hr/urn:nbn:hr:237:781222

N. Adžaga, A. Filipin and Z. Franušić, "On the extensions of the Diophantine triples in Gaussian integers", Monatshefte für Mathematik, vol. 197(4), pp. 535-563, 2022. [Online]. Available at: https://urn.nsk.hr/urn:nbn:hr:237:781222. [Accessed: 04 November 2024]

Metadata
Title (english)On the extensions of the Diophantine triples in Gaussian integers
AuthorNikola Adžaga
AuthorAlan Filipin
AuthorZrinka Franušić
Author's institutionUniversity of Zagreb
Faculty of Civil Engineering
Scientific / art field,
discipline and subdiscipline		TECHNICAL SCIENCES
Civil Engineering
Abstract (english)
A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple {; ; k−1, k+1, 16k3−4k}; ; in Gaussian integers Z[i] to a Diophantine quadruple. Similar one- parameter family, {; ; k − 1, k + 1, 4k}; ; , was studied in [9], where it was shown that the extension to a Diophantine quadruple is unique (with an element 16k3 −4k). The family of the triples of the same form {; ; k−1, k+1, 16k^3 −4k}; ; was studied in rational integers in [6]. It appeared as a special case while solving the extensibility problem of Diophantine pair {; ; k − 1, k + 1}; ; , in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between k + 1 and 16k^3 − 4k is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.
Keywords (english)
Languageenglish
Publication typeScientific paper - Original scientific paper
Publication statusPublished
Peer reviewPeer review - international
Publication versionPublished version
Journal titleMonatshefte für Mathematik
Numberingvol. 197(4), pp. 535-563
e-ISSN1436-5081
URN:NBNurn:nbn:hr:237:781222
Publication2022
Type of resourceText
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2024-09-05 05:10:53