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Figural numbers

A figural number (also called fiigurate numbers)  is a number that can be represented by a regular geometrical arrangement of equally spaced points (pebbles).

If the arrangement forms a planar regular polygon, the number is called a polygonal number. To these group belong triangular, square, pentagonal, and hexagonal numbers, respectively.

Arithmetically they can be defined as the partial sums of arithmetical progressions

1 2 3 4 ... n ... 1 3 5 ... 10 ... 3 n - 2 ... 1 5 9 13 ... 4 n - 3 ...

Thus the typeset structureth typeset structure-gonal number typeset structure is the sum of the first typeset structure terms of the arithmetic progression with first term 1 and common difference typeset structure

p _ n^(r) = n + (r - 2) n(n - 1)/2 .(1)

To compute the typeset structureth typeset structure-gonal number go to .

To compute a segment of elements of the sequence of i-gonal number go to .

Triangular numbers can be considered as an additive analog of the factorial.

Figurate numbers can also form other planar shapes such as centered polygons, gnomons ( L-shapes), etc.

A figural number corresponding to a spatial configuration of points which form a pyramid with typeset structure-sided regular polygon bases can are called generalized pyramidal numbers. The typeset structureth typeset structure-sided pyramidal number typeset structure can be also defined as the partial sum of the first typeset structure terms of  the sequence typeset structure-gonal numbers.  The formula for this sum was found by Roman geometers Epaphroditus and Vitrius Rufus (students of Heron of Alexandria) around 150

P _ n^(k) = 1/6 (n + 1) (2 p _ n^(k) + n) .(2)

This relation can be considered as a generalization of the well-known relation

Underoverscript[∑, k = 1, arg3] i^2 = (n(n + 1) (n + 2))/6 .

For typeset structure we get tetrahedral numbers, for typeset structure the square pyramidal numbers, for typeset structure pentagonal ones, etc.

The pyramidal numbers can also be generalized to four or higher dimensions.

Polygonal numbers were studied as long as Pythagoras (525 BC). Nicomachus of Gerasa mentions them in his Introductio Arithmeticae (100 AD). Diophantus wrote a treatise on them (250AD), which was translated(and commented)  in Latin by Bachet. In 1638 Fermat conjectured that every positive integer is a sum of 3 triangular numbers, 4 square numbers, 5 pentagonal numbers, and so on. In 1796 Gauß proved the case of the triangular numbers . The proof is published in his Disquisitiones Arithmeticae (1801). In 1770 Lagrange proved the case of the square numbers:

Lagrange’s Four-Square Theorem. Every positive integer is a sum of four integer squares.

  Cauchy proved Fermat’s conjecture in 1813.

Cauchy’s Polygonal Number Theorem: Let typeset structure and typeset structure be positive integers. Then there exist non-negative integers typeset structure and typeset structure, such that

n = (t + r (t^2 - t)/2) + (u + r (u^2 - u)/2) + (v + r (v^2 - v)/2) + (w + r (w^2 - w)/2) + m ,

where typeset structure.

The result says that at most 4 of the polygonal numbers have to be typeset structure.

In 1830 Legendre proved that four or five polygonal numbers are enough for all sufficiently large typeset structure.

Legendre’s Polygonal Number Theorem:  Let

typeset structure. If typeset structure is even, then every integer typeset structure is the sum of five typeset structure-gonal numbers, at least one of which is either 0 or 1. If typeset structure is odd, then every integer typeset structure is the sum of four typeset structure-gonal numbers.

Cite this web-page as:

Štefan Porubský: Figural Numbers.


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