An implementation of Runge's method for Diophantine equations

The Nagell-Ljunggren equation via Runge's method

The Diophantine equation (x^n-1)/(x-1)=y^q has four known solutions in integers x, y, q and n with |x|, |y|, q > 1 and n > 2. Whilst we expect that there are, in fact, no more solutions, such a result is well beyond current technology. In this paper, we prove that if (x,y,n,q) is a solution to this equation, then n has three or fewer prime divisors, counted with multiplicity. This improves a result of Bugeaud and Mihailescu.

逆向工程的丢番图方程

Reverse engineered Diophantine equations

本文回答了Samir Siksek提出的问题，利用Mihăilescu定理、Runge方法和Fermat大定理，为任意有限完美整数幂集合构造多项式，使其包含的完美整数幂恰好属于该集合，实现了丢番图方程的逆向工程。

求解丢番图方程 ay^2=b(x^k-c^k) 的若干方法

Some Methods to Solve the Diophantine Equation ay^2=b(x^k-c^k)

本文针对丢番图方程 ay^2=b(x^k-c^k) 的不同 k 情形提出便于计算机实现的求解方法，通过降次提高搜索效率，并给出方程不可解的充分条件。