Computing all integer solutions of a genus 1 equation

Abstract

Abstract. The elliptic logarithm method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f (u, v) = 0, where f ∈ Z [u, v] is irreducible over Q, defines a curve of genus 1, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree 5 and degree 9.

Combinatorial numbers in binary recurrences

“…There is an efficient method to determine the solutions of such equations developed by Gebel, Pethő and Zimmer [12] and independently by Stroeker and Tzanakis [32]. Later, the method was further improved and generalized by Stroeker and Tzanakis (see [33] and the references given there). Moreover, the program package Magma contains procedures (based on these results) to resolve such equations.…”

A public key cryptosystem based on diophantine equations of degree increasing type

“…However, there are no known general algorithms to solve such diophantine equations in polynomial time. For instance, in [20], it was proved that the problem for determining whether there are positive integer solutions for [6,26,30]. Otherwise, in special cases there are some algorithms to find all integral points [3,4].…”

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

“…In recent years, the determination of integral points on elliptic curves has been an interesting problem in number theory and arithmetic algebraic geometry, and many advanced methods have been used to solve this problem (see [1], [5], [6]). …”