A New Polygonal Number Theorem

The polygonal number theorem of Fermat, Cauchy, and Legendre has served as one of the leading results in the history of additive number theory. It states that every positive integer can be written as the sum of m m-gonal numbers, and Legendre improved this to four or five m-gonal numbers for sufficiently large integers. A variation of this problem is to determine the minimal amount of m-gonal numbers needed in order to represent all integers as the sum and difference of these elements infinitely many different ways. Fortunately, a full solution is provided to this problem as the following result.