Some experiments with Ramanujan-Nagell type Diophantine equations

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