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## EUREKA!NUM=Δ+Δ+Δ

The above words and symbols are one entry in Carl Frederik Gauss's diary in the early-19th-century. Any idea what they mean?

The Fermat Polygonal Number Theorem states every positive integer is a sum of at most n n-gonal numbers (e.g. 3-gonals are triangular numbers, 4-gonals are square numbers, etc.). In the case of 4-gonal numbers, 7=1+1+1+4, 8=4+4, etc.).

Fermat is credited with the big idea, though he only stated it without proof. He did promise to share a proof later...but it never appeared (sound familar?).

In 1770, Joseph Lagrange proved the 2-gonal or square case, showing that every positive number can be represented as a sum of at most four squares.

In 1796, Carl Gauss proved the 3-gonal or triangular case (e.g. 7=1+6, 8 = 1+3+6, etc.). To document or commemorate his discovery, Gausee wrote "ΕUREΚΑ! NUM = Δ + Δ + Δ" in his diary, then later published his proof in his Disquisitiones Arithmeticae.

In 1813, Augustin-Louis Cauchy proved the theorem for the general case...n n-gonal numbers. However, the theorem is often known as the EUREKA theorem after Gauss.