2014/09/29 by Maciej Ulas, Ulas, Maciej
Mathematics · #11D41 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D41
paper · pdf · doi:10.48550/arxiv.1409.8132
14 pages, to appear in Galsnik Matematicki
arxiv created 2014/09/29 · openalex publication_date 2014/09/29 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,B∈\Z (thus A, B are not necessarily positive) and k∈\Z≥ 2 are given integers. In particular, we prove that for each k there exists an infinite set \calS containing pairs of integers (A, B) such that for each (A,B)∈ \calS 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. %For example the equation y2=130⋅ 3n+5550606 has exactly five solutions with n=0, 6, 11, 15, 16. We also find new examples of equations x2=A2n+B having six solutions in positive integers, e.g. the following Diophantine equations has 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.