EUREKA! NUM=Δ+Δ+Δ
The above words and symbols are one entry in Carl Frederik Gauss's diary in the early19thcentury. Any idea what they mean?
The Fermat Polygonal Number Theorem states every positive integer is a sum of at most n ngonal numbers (e.g. 3gonals are triangular numbers, 4gonals are square numbers, etc.). In the case of 4gonal 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 2gonal 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 3gonal 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, AugustinLouis Cauchy proved the theorem for the general case...n ngonal numbers. However, the theorem is often known as the EUREKA theorem after Gauss.
