KnowraFermat polygonal number theoremFermat polygonal number theoremFor every integer n ≥ 3, each nonnegative integer is a sum of at most n n-gonal numbers, including zero.BriefConnectPolygonal number: A number in the sequence represented by a regular polygonal arrangement of dots. The theorem asserts that every nonnegative integer is a bounded sum of these numbers.Pierre de Fermat: A seventeenth-century French mathematician known for work in number theory and geometry. He stated the polygonal-number claim as a conjecture.Regular polygon: A polygon with equal sides and equal interior angles. The geometric shapes give polygonal numbers their names and dot-arrangement interpretation.Gauss's Eureka theorem: The theorem that every nonnegative integer is a sum of three triangular numbers. It is the n = 3 instance and the case Gauss proved in 1796.Triangular number: A number of the form k(k+1)/2 for a nonnegative integer k. The three-sided case is Gauss’s theorem that every nonnegative integer is a sum of three triangular numbers.Carl Friedrich Gauss: A German mathematician whose work shaped number theory, geometry, and mathematical physics. In 1796 he proved the triangular-number case and recorded the result in his diary.Figurate number: A number represented by a regular geometric arrangement of dots. Polygonal numbers are the two-dimensional members of this broader family.Square number: An integer of the form k² for an integer k. The n = 4 instance says every nonnegative integer is a sum of at most four squares.Quadratic form: A homogeneous polynomial of degree two in several variables. Polygonal-number representations become quadratic-form equations after multiplying by a fixed denominator.Fermat's conjecture on polygonal numbers: Fermat’s conjecture that every positive integer is a sum of at most n n-gonal numbers. The theorem is the proved form of this conjecture, extended to include zero.Show all 21Linked from 2 pagesAdditive number theoryRelated: It extends representation by squares to broader families of polygonal numbers.Legendre's three-square theoremRelated: It belongs to the broader history of characterizing integers represented by sums of special numbers.