Some experiments with Ramanujan-Nagell type Diophantine equations

Abstract

Abstract. Stiller proved that the Diophantine equation x 2 + 119 = 15 · 2 n has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type x 2 = Ak n + 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 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 x 2 = Ak n + B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x 2 = Ak n + B with k > 2, each containing five solutions in non-negative integers. We also find new examples of equations x 2 = A2 n + B having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions:x 2 = 57 · 2 n + 117440512 n = 0, 14, 16, 20, 24, 25, x 2 = 165 · 2 n + 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.

On the Ramanujan-Nagell type Diophantine equation x2+Akn=B

“…In this paper, we prove that the Ramanujan-Nagell type Diophantine equation x 2 + Ak n = B has at most three nonnegative integer solutions (x, n) for A = 1, 2, 4, k an odd prime and B a positive integer. Therefore, we partially confirm two conjectures of Ulas from [23]. …”

Persamaan Diophantine Tipe Ramanujan-Nagell x2 = yn + 2185

“…Adapun usaha untuk mencari solusi persamaan (1) terus dilakukan, salah satu hasil penelitian terbaru telah dipublikasikan oleh Maciej Ulas pada tahun 2014. Dalam tulisannya [4], Ulas mengajukan sebuah konjektur berikut ini:…”

On two Diophantine equations of Ramanujan-Nagell type

“…One can see for examples [1]- [21]. One aspect of the study of equation (1.2) is to determine the integer solutions (x, k, n).…”