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This paper serves as a collection of research(done by other mathematicians) on polygonal numbers as well as their corresponding theorem proof summary proved by other mathematicians. This paper mainly reviews triangular numbers, square numbers, Gauss's Triangular Number Theorem, Lagrange’s Four-Square Theorem, Cauchy’s Polygonal Number Theorem, Fermat’s Polygonal Number Theorem. This composition also briefly reviews topics regarding Waring’s problem and sum of three cubes. The conclusion is that mathematicians have proved that any positive integer can be decomposed as a fixed number of “2-dimensional” polygonal numbers, such as triangular, square, pentagonal numbers and so on. Later studies related to polygonal numbers will focus on “higher dimensional” polygonal numbers, such as cubic numbers.
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