# 基于三维递推算法的牛顿二项式定理研究Newton’s Binomial Theorem Based on Three-Dimensional Recursive Algorithms

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The recursive algorithm of Newton’s binomial expansion calculates directly (x + y)k−1 without calculating (x + y)k, where the exponential term k is a “two-dimensional” rational number (one-dimensional molecule and denominator) or a negative number. To solve the problem of calculating the coefficients of Newton’s binomial expansion terms, this paper presents a new “three-dimensional” recursive algorithm for splitting functions with three variables. This method transforms the step-by-step recursive process into the splitting calculation of integers (or rational numbers) and then directly calculates the coefficients of arbitrary expansion terms. At the same time, it proves and explains that the essence of the coefficients of Newton’s binomial expansion terms is the ratio of splitting functions. On the other hand, when k is not a positive integer, the polynomial rings given by Newton’s binomial expansion are all in the form of infinite series. In addition to polynomial rings in infinite series form, there are polynomial rings in finite sequence and form in the expansions given in this paper. Therefore, the new algorithm can be used to further study the prime elements in the Gauss integer ring.

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