Fermat's Polygonal Number Theorem
In 1638, Fermat 
proposed that every Positive Integer is a sum of at most threeTriangular Numbers, four Square Numbers, five PentagonalNumbers, and 
-Polygonal Numbers. Fermat claimed to have a proof ofthis result, although Fermat's proof has never been found. Gauß proved the triangular case, and noted the eventin his diary on July 10, 1796, with the notation
This case is equivalent to the statement that every number of the form
is a sum of three Odd Squares (Duke 1997). More specifically, a number is a sum of three Squares Iff it is not of the form 
for 
, as first proved by Legendre in 1798.Euler was unable to prove the square case of Fermat's theorem, but he left partial results which were subsequentlyused by Lagrange. The square case was finally proved by Jacobi and independently by Lagrange in 1772. Itis therefore sometimes known as Cauchy proved the proposition in itsentirety.
References
Cassels, J. W. S. Rational Quadratic Forms. New York: Academic Press, 1978. Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. ``The Primary Pretenders.'' Acta Arith. 78, 307-313, 1997. Duke, W. ``Some Old Problems and New Results about Quadratic Forms.'' Not. Amer. Math. Soc. 44, 190-196, 1997. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 143-144, 1993. Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 91, 1984.
- Nephroid Involute
- Nerve
- Nested Hypothesis
- Nested Radical
- Net
- Net (Polyhedron)
- Netto's Conjecture
- Network
- Neuberg Circles
- Neumann Algebra
- Neumann Boundary Conditions
- Neumann Function
- Neumann Polynomial
- Neumann Series (Bessel Function)
- Neumann Series (Integral Equation)
- Neusis Construction
- Neville's Algorithm
- Neville Theta Function
- Newcomb's Paradox
- Newman-Conway Sequence
- Newton-Cotes Formulas
- Newtonian Form
- Newton Number
- Newton-Raphson Fractal
- Newton-Raphson Method