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Glasnik matematički, Vol. 49 No. 2, 2014.

Izvorni znanstveni članak
https://doi.org/10.3336/gm.49.2.04

Some experiments with Ramanujan-Nagell type Diophantine equations

Maciej Ulas ; Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6, 30 - 348 Kraków, Poland

Puni tekst: engleski, pdf (164 KB) str. 287-302 preuzimanja: 21* citiraj
APA 6th Edition
Ulas, M. (2014). Some experiments with Ramanujan-Nagell type Diophantine equations. Glasnik matematički, 49 (2), 287-302. https://doi.org/10.3336/gm.49.2.04
MLA 8th Edition
Ulas, Maciej. "Some experiments with Ramanujan-Nagell type Diophantine equations." Glasnik matematički, vol. 49, br. 2, 2014, str. 287-302. https://doi.org/10.3336/gm.49.2.04. Citirano 18.02.2019.
Chicago 17th Edition
Ulas, Maciej. "Some experiments with Ramanujan-Nagell type Diophantine equations." Glasnik matematički 49, br. 2 (2014): 287-302. https://doi.org/10.3336/gm.49.2.04
Harvard
Ulas, M. (2014). 'Some experiments with Ramanujan-Nagell type Diophantine equations', Glasnik matematički, 49(2), str. 287-302. doi: https://doi.org/10.3336/gm.49.2.04
Vancouver
Ulas M. Some experiments with Ramanujan-Nagell type Diophantine equations. Glasnik matematički [Internet]. 2014 [pristupljeno 18.02.2019.];49(2):287-302. doi: https://doi.org/10.3336/gm.49.2.04
IEEE
M. Ulas, "Some experiments with Ramanujan-Nagell type Diophantine equations", Glasnik matematički, vol.49, br. 2, str. 287-302, 2014. [Online]. doi: https://doi.org/10.3336/gm.49.2.04

Sažetak
Stiller proved that the Diophantine equation x2+119=15 · 2n has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type x2=Akn+B with many solutions. Here, A,Bℤ (thus A, B are not necessarily positive) and kℤ ≥ 2 are given integers. In particular, we prove that for each k there exists an infinite set S containing pairs of integers (A, B) such that for each (A,B) S we have gcd(A,B) is square-free and the Diophantine equation x2=Akn+B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x2=Akn+B with k>2, each containing five solutions in non-negative integers. We also find new examples of equations x2=A2n+B having six solutions in positive integers, e.g. the following Diophantine equations have exactly six solutions:



x2= 57· 2n+117440512, n=0 , 14 , 16, 20, 24, 25,
x2= 165· 2n+26404, n=0 , 5 , 7, 8, 10, 12.

Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.

Ključne riječi
Diophantine equation; Ramanujan-Nagell equation

Hrčak ID: 130884

URI
https://hrcak.srce.hr/130884

Posjeta: 112 *