A brief history of partitions of numbers, partition functions and their modern applications

'Number rules the universe.' The Pythagoras 'If you wish to forsee the future of mathematics our course is to study the history and present conditions of the science.' Henri Poincaré 'The primary source (Urqell) of all mathematics are integers.' Hermann Minkowski This p...

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Vydáno v:International journal of mathematical education in science and technology Ročník 47; číslo 3; s. 329 - 355
Hlavní autor: Debnath, Lokenath
Médium: Journal Article
Jazyk:angličtina
Vydáno: London Taylor & Francis 02.04.2016
Taylor & Francis, Ltd
Taylor & Francis Ltd
Témata:
1100
applications to statistical physics
asymptotic formulas of partition functions
Basic Skills
Equations (Mathematics)
generating functions
History
Mathematical Applications
Mathematical Formulas
Mathematics
Mathematics Education
Mathematics Instruction
Number Concepts
Numbers
partition functions
partitions of numbers
Statistics
ISSN:0020-739X, 1464-5211
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Shrnutí:'Number rules the universe.' The Pythagoras 'If you wish to forsee the future of mathematics our course is to study the history and present conditions of the science.' Henri Poincaré 'The primary source (Urqell) of all mathematics are integers.' Hermann Minkowski 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.
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ISSN:0020-739X
1464-5211
DOI:10.1080/0020739X.2015.1071441