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Fermat's polygonal number theorem states that every positive integer can be written as the sum of at most -gon numbers. This implies that a positive integer can be written as the sum of at most 3 triangular numbers, 4 square numbers, and so on.
- (triangular numbers)
- (square numbers)
- (pentagonal numbers)
The case for was proved by Gauss and the proof is given below.
From Lagrange's three-square theorem, we know that a number of the form cannot be represented as the sum of 3 square numbers. This implies that numbers of the form can be written as the sum of 3 squares.
The squares mod 8 are and
For to be the sum of three squares, the squares need to be congruent to
All triangular numbers are of the form Thus, can be written as the sum of three triangular numbers.