**ERIC Number:**EJ1091949

**Record Type:**Journal

**Publication Date:**2016

**Pages:**27

**Abstractor:**As Provided

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

**EISSN:**N/A

A Brief History of Partitions of Numbers, Partition Functions and Their Modern Applications

Debnath, Lokenath

International Journal of Mathematical Education in Science and Technology, v47 n3 p329-355 2016

This paper is written to commemorate the centennial anniversary of the Mathematical Association of America. It deals with a short history of different kinds of natural numbers including triangular, square, pentagonal, hexagonal and "k"-gonal numbers, and their simple properties and their geometrical representations. Included are Euclid's and Pythagorean's main contributions to elementary number theory with the main contents of the "Euclid Elements" of the 13-volume masterpiece of mathematical work. This is followed by Euler's new discovery of the additive number theory based on partitions of numbers. Special attention is given to many examples, Euler's theorems on partitions of numbers with geometrical representations of Ferrers' graphs, Young's diagrams, Lagrange's four-square theorem and the celebrated Waring problem. Included are Euler's generating functions for the partitions of numbers, Euler's pentagonal number theorem, Gauss' triangular and square number theorems and the Jacobi triple product identity. Applications of the theory of partitions of numbers to different statistics such as the Bose-Einstein, Fermi-Dirac, Gentile, and Maxwell-Boltzmann statistics are briefly discussed. Special attention is given to pedagogical information through historical approach to number theory so that students and teachers at the school, college and university levels can become familiar with the basic concepts of partitions of numbers, partition functions and their modern applications, and can pursue advanced study and research in analytical and computational number theory.

Descriptors: Mathematics, Mathematics Instruction, Mathematical Applications, Numbers, Statistics, Mathematical Formulas, Equations (Mathematics), History

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Grant or Contract Numbers:**N/A