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In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon [1]: 2-3 . These are one type of 2-dimensional figurate numbers . Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong , triangular , and square numbers ...
Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles. The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem.
The n-th centered k-gonal number is equal to the n-th regular k-gonal number plus (n-1) 2. Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:
The maximum number of pieces, p obtainable with n straight cuts is the n-th triangular number plus one, forming the lazy caterer's sequence (OEIS A000124) The maximum number p of pieces that can be created with a given number of cuts n (where n ≥ 0) is given by the formula
If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.
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.
Has proofs of Lagrange's theorem, the polygonal number theorem, Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent N as the sum of s kth powers. Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics (1933) (ISBN 0-691-02351-4). Has a proof of the ...
Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the n th centered k-gonal number is obtained by the formula = + where T is a triangular number. The positive difference of two triangular numbers is a trapezoidal number.