Power Savings for Counting Solutions to Polynomial-Factorial Equations
Advances in mathematics(2023)
摘要
Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n≤ N to n! = P(x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N). The proof uses techniques of Diophantine and Padé approximation.
更多查看译文
关键词
Polynomial-factorial equation,Simultaneous rational approximation,Pad? approximation
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要