In mathematics, the Fermat polygonal number theorem states: every positive integer is a sum of at most n n-polygonal numbers. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...

An example of triangular number case would be 17 = 10 + 6 + 1.

A well-known special case of this is Lagrange's four-square theorem, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1. Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ...

Joseph Louis Lagrange proved the square case in 1770 and Gauss proved the triangular case in 1796, but the theorem was not resolved until it was finally proven by Cauchy in 1813. Nathanson's proof (see the references) is based on the following lemma due to Cauchy: Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

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

• Lagrange's four-square theorem
• Polygonal number
• Pollock octahedral numbers conjecture

Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...

References

• Eric W. Weisstein. "Fermat's Polygonal Number Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html
• Nathanson, M. B. "A Short Proof of Cauchy's Polygonal Number Theorem." Proc. Amer. Math. Soc. Vol. 99, No. 1, 22-24, (Jan. 1987).

External links

• Eric W. Weisstein, Fermat's Polygonal Number Theorem at MathWorld.