Methods to Find Integer Points on the Elliptic Curve of Factorizable Polynomial Equation

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)(𝑥2 + 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.