求可因式分解多项式方程的椭圆曲线上的整数点的方法
Methods to Find Integer Points on the Elliptic Curve of Factorizable Polynomial Equation
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中文总结 AI 辅助
针对两类可因式分解的椭圆曲线，利用初等数论将其整数点问题转化为Pell方程求解，提出通用高效方法并给出不可解判定条件。
中文摘要 AI 辅助
作为一个有价值的理论和应用问题，研究了两类椭圆曲线y^2=(x-a)(x-b)(x-c)和y^2=(x-a)(x^2+ax+b)上的整数点。通过使用初等数论方法，将原方程的解归结为联立Pell方程或广义Pell方程的解，这可以提高计算机搜索整数解的效率。提出了求方程整数解的有效方法，便于算法编程实现。同时，方法的推导得出了两个充分条件，可分别判定这两类方程的可解性。所提出的方法适用于系数未定的方程，比求解具有特定已知系数的方程更具一般性，且更适合计算机编程实现。
英文摘要
As a valuable theoretical and application problem, the integer points on two types of elliptic curve y^2=(x-a)(x-b)(x-c) and y^2 = (x-a)(x^2+ax+b) are studied. By using elementary number theory method, the solution of the original equations is reduced to the solution of simultaneous Pell equations or generalized Pell equation, which can improve the efficiency in searching integer solutions by computer. Effective methods are proposed to find the integer solutions of the equations, which are convenient for algorithm programming implementation. At the same time, the derivation of the methods leads to two sufficient conditions, which can determine the unsolvability of these two types of equation respectively. The methods proposed are for equations with undetermined coefficients, which are more general than the solutions of the equations with specific known coefficients, and more suitable for computer programming implementation.
发表机构
- Technical University of Sofia（索菲亚技术大学）
- International Research Institute of Biomedicine, Natural Sciences and Technology（生物医学、自然科学与技术国际研究所）
- Qingdao University（青岛大学）
机构由 AI 辅助整理，请以论文原文为准。
