论文标题
为多项式方程计算解决方案的动力节省
Power savings for counting solutions to polynomial-factorial equations
论文作者
论文摘要
令$ p $为多项式，具有整数系数和至少两个。我们证明了整数解决方案$ n \ leq n $ to $ n的上限！ = P（x）$，可以在微不足道的界限上节省电力。特别是，这适用于Brocard和Ramanujan的百年历史。以前的最佳结果是解决方案的数量为$ O（n）$。该证明使用二芬太汀和帕德近似的技术。
Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ 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.
