高维线性分拆理论及其应用——Ramsey数R(5,5;2)=47的注记

The Theory of Higher Dimensional Linear Partitions and Its Application—Note on Ramsey Number R(5,5;2)=47

摘要: Euler通过展开一元表达式(1+x^α z)(1+x^β z)(1+x^γ z)⋯的研究，创立了数的分拆这个数学分支。本文推广表达式到二元形式(1+x^a y^α z)(1+x^b y^β z)(1+x^c y^γ z)⋯,并建立了相应的二元分拆的递推规则，由此得到了许多有趣的性质，特别是建立了圆周率 π/4的莱布尼茨公式和圆周率 π²/6的欧拉公式与二元分拆函数的联系。二元分拆本质上属于立方体分拆或双二元函数分拆，同时它也是哥德尔数推广到哥德尔数组的一种表达形式。事实上, 二元分拆理论也为求解拉姆塞数R(5,5；2)=47找到了一种新方法。

Abstract: Leonhard Euler established a new mathematical branch: partition, from his study on generating functions: (1+x^α z)(1+x^β z)(1+x^γ z)⋯. In this work, we propose several new recurrence relations from dual-binary function of solid partitions and General Gödel numbers, based on observations of (1+x^a y^α z)(1+x^b y^β z)(1+x^c y^γ z)⋯ inspired by Euler’s work. Furthermore connections are derived from this, especially of Leibniz formula for π/4 and Euler formula for π²/6 , from binary pertition formulas. Binary partition is an implementation of cubic partition or dual-binary partitions, which is as well as a expression from Gödel numbers to Gödel numbers array. Actually, we present a new sense to get Ramsey number R(5, 5; 2) = 47 in this paper.