Fermat polygonal number theorem
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
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]
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.
See also
Notes
- ^ a b c Heath (1910).
- ISBN 0-486-41150-8.
- S2CID 122203472.
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.
- MR 0866422.
- Nathanson, Melvyn B. (1996), Additive Number Theory The Classical Bases, Berlin: ISBN 978-0-387-94656-6. Has proofs of Lagrange's theorem and the polygonal number theorem.