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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. Joseph Louis Lagrange proved the square case in 1770 and Gauss proved the triangular case in 1796, but the theorem was not resolved until it was finally proven by Cauchy in 1813. For odd positive integers a and b such that b2 4a and 3a b2 + 2b + 4 we can find nonnegative integers s,t,u and v such that a = s2 + t2 + u2 + v2 and b = s + t + u + v.