Fermat polygonal number theorem
Every positive integer is a sum of at most n n-gonal numbers
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Examples
Three such representations of the number 17, for example, are shown below:
- 17 = 10 + 6 + 1 (triangular numbers)
- 17 = 16 + 1 (square numbers)
- 17 = 12 + 5 (pentagonal numbers).
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History
The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813. The proof of Nathanson (1987) is based on the following lemma due to Cauchy:
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.