Glasnik Matematicki, Vol. 49, No. 2 (2014), 287-302.

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
e-mail: maciej.ulas@uj.edu.pl

Abstract.   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.

2010 Mathematics Subject Classification.   11D41.

Key words and phrases.   Diophantine equation, Ramanujan-Nagell equation.

Full text (PDF) (free access)

DOI: 10.3336/gm.49.2.04

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