Fermat polygonal number theorem

Last updated April 18, 2023

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most $n$ $n$ -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. That is, the $n$ -gonal numbers form an additive basis of order $n$ .

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).

History

Gauss's diary entry related to sum of triangular numbers (1796)

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.^[1] 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.^[1] Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! num = Δ + Δ + Δ",^[2] and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem.^[3] The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813.^[1] The proof of Nathanson (1987) is based on the following lemma due to Cauchy:

For odd positive integers $a$ and $b$ such that $b 2 < 4 a$ and $3 a < b 2 + 2 b + 4$ we can find nonnegative integers $s$ , $t$ , $u$ , and $v$ such that $a = s 2 + t 2 + u 2 + v 2$ and $b = s + t + u + v$ .

Notes

1 2 3 Heath (1910).
↑ Bell, Eric Temple (1956), "Gauss, the Prince of Mathematicians", in Newman, James R. (ed.), The World of Mathematics, vol. I, Simon & Schuster, pp. 295–339. Dover reprint, 2000, ISBN 0-486-41150-8.
↑ Ono, Ken; Robins, Sinai; Wahl, Patrick T. (1995), "On the representation of integers as sums of triangular numbers", Aequationes Mathematicae , 50 (1–2): 73–94, doi:10.1007/BF01831114, MR 1336863, S2CID 122203472 .

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References

Weisstein, Eric W. "Fermat's Polygonal Number Theorem". MathWorld .
Heath, Sir Thomas Little (1910), Diophantus of Alexandria; a study in the history of Greek algebra, Cambridge University Press, p. 188.
Nathanson, Melvyn B. (1987), "A short proof of Cauchy's polygonal number theorem", Proceedings of the American Mathematical Society, 99 (1): 22–24, doi:10.2307/2046263, JSTOR 2046263, MR 0866422 .
Nathanson, Melvyn B. (1996), Additive Number Theory The Classical Bases, Berlin: Springer, ISBN 978-0-387-94656-6 . Has proofs of Lagrange's theorem and the polygonal number theorem.

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[heath-1] 1 2 3 Heath (1910).

[2] Bell, Eric Temple (1956), "Gauss, the Prince of Mathematicians", in Newman, James R. (ed.), The World of Mathematics, vol. I, Simon & Schuster, pp. 295–339. Dover reprint, 2000, ISBN 0-486-41150-8.

[3] Ono, Ken; Robins, Sinai; Wahl, Patrick T. (1995), "On the representation of integers as sums of triangular numbers", Aequationes Mathematicae , 50 (1–2): 73–94, doi:10.1007/BF01831114, MR 1336863, S2CID 122203472 .

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v t e Pierre de Fermat
Work	Fermat's Last Theorem Fermat number Fermat's principle Fermat's little theorem Fermat polygonal number theorem Fermat pseudoprime Fermat point Fermat's theorem (stationary points) Fermat's theorem on sums of two squares Fermat's spiral Fermat's right triangle theorem
Related	List of things named after Pierre de Fermat Wiles's proof of Fermat's Last Theorem Fermat's Last Theorem in fiction Fermat Prize Fermat's Last Tango (2000 musical) Fermat's Last Theorem (popular science book)