Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics

TL;DR

This paper develops highly accurate, easy-to-use approximation formulas for the partition number p(n) by fitting parameters in Hardy-Ramanujan's formula, enabling quick calculations without programming.

Contribution

It introduces revised elementary estimation formulas for p(n) using data fitting of parameters, improving accuracy and simplicity over existing methods.

Findings

New approximation formulas achieve high accuracy for p(n)

Methods allow calculation with a pocket calculator without programming

Fitting techniques can be applied to other complex data fitting problems

Abstract

Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either inconvenient for most people (not majored in math) who do not want do write programs, or unsatisfying in accuracy. By bringing in two parameters in the Hardy-Ramanujan's Asymptotic formula and fitting the data of the two parameters by least square method, iteration method and some other special designed methods, several revised elementary estimation formulae with high accuracy for p(n) are obtained. With these estimation formulae, the approximate value of p(n) can be calculated by a pocket calculator without programming function. The main difficulty is that the usual methods to fit the data of the two parameters by an elementary function is defective here. These method could be used in finding the fitting functions of some other complex data.