The number partition problem is a well-known problem, which is one of 21 Karp's NP-complete problems \cite{karp}. The partition function is a boolean function that is equivalent to the number partition problem with number range restricted. To fully understand the computational complexity of the number partition problem and the partition function is quite important and hard. People speculate that we need new tools and methods \cite{aaronson} for such problem. In our recent research on the universal learning machine \cite{paper5, paper8}, we developed some tools, namely, fitting extremum, proper sampling set, boolean function with parameters (used in trial-and-error fashion). We found that these tools could be applied to the partition function. In this article, we discuss the set up of the partition function, properties of the partition function, and the tools to be used. This approach leads us to prove that the lower bound of the computational complexity of partition function, as well as the lower bound of the computational complexity of the number partition problem, is exponential to the size of problem. This implies: {\bf P} $\ne$ {\bf NP} \cite{cook}.

翻译：数字分区问题是一个众所周知的问题, 这是 21 个 Karp 的 NP- 完整的问题 \ cite{karp} 。 分区函数是一个布尔函数, 与数字分区问题相同, 数量范围受限制 。 要完全理解数字分区问题和分区功能的计算复杂性是相当重要和困难的。 人们猜测, 这个问题我们需要新的工具和方法 \ cite { aaronson} 。 在我们最近对通用的 学习机器 \ cite{ paper5, paper8} 的研究中, 我们开发了一些工具, 即 安装 extremum, 正确的取样装置, 布林函数与参数( 用于试用和试用时) 。 我们发现这些工具可以应用到分区函数 。 在此文章中, 我们讨论分区函数的设置、 分区功能的属性以及要使用的工具 。 这个方法可以证明, 分区功能的计算复杂性较低, 以及数字分区问题的计算复杂性的较低约束, 与问题大小是指数 $( =\\\ p) 。