Triangular number

The first six triangular numbers (not starting with T0)

Equilateral triangle.

- Triangular number
The first six triangular numbers (not starting with T0)

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The first six triangular numbers (not starting with T0)

Figurate number

The first six triangular numbers (not starting with T0)

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers).

One interpretation of Plato's number, 33 + 43 + 53 = 63

Cube (algebra)

[[Image:CubeChart.svg|thumb|280px|

[[Image:CubeChart.svg|thumb|280px|

One interpretation of Plato's number, 33 + 43 + 53 = 63

The sum of the first n cubes is the nth triangle number squared:

Square number 16 as sum of gnomons.

Square number

Integer that is the square of an integer; in other words, it is the product of some integer with itself.

Integer that is the square of an integer; in other words, it is the product of some integer with itself.

Square number 16 as sum of gnomons.
Proof without words for the sum of odd numbers theorem

. If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers).

Archimedes exclaiming Eureka. In his excitement, he forgets to dress and runs nude in the streets straight out of his bath

Eureka (word)

Interjection used to celebrate a discovery or invention.

Interjection used to celebrate a discovery or invention.

Archimedes exclaiming Eureka. In his excitement, he forgets to dress and runs nude in the streets straight out of his bath
16th-century illustration of Archimedes in the bath, with Hiero's crown at bottom right
The Seal of California, featuring the word "EUREKA" above the spear of the goddess Minerva, from 1870
Gauss's diary entry related to sum of triangular numbers (1796)

Another mathematician, Carl Friedrich Gauss, echoed Archimedes when in 1796 he wrote in his diary, "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", referring to his discovery that any positive integer could be expressed as the sum of at most three triangular numbers.

An s-gonal number can be decomposed into s−2 triangular numbers and a natural number

Fermat polygonal number theorem

Sum of at most n n-gonal numbers.

Sum of at most n n-gonal numbers.

An s-gonal number can be decomposed into s−2 triangular numbers and a natural number

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.

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

4000 (number)

Natural number following 3999 and preceding 4001.

Natural number following 3999 and preceding 4001.

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

4005 – triangular number

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Ramanujan–Nagell equation

Equation between a square number and a number that is seven less than a power of two.

Equation between a square number and a number that is seven less than a power of two.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent:

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

190 (number)

Natural number following 189 and preceding 191.

Natural number following 189 and preceding 191.

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties.

The 24-hour tower clock in Venice, using J as a symbol for 1

1

Number and a numerical digit used to represent that number in numerals.

Number and a numerical digit used to represent that number in numerals.

The 24-hour tower clock in Venice, using J as a symbol for 1
This Woodstock typewriter from the 1940s lacks a separate key for the numeral 1.
Hoefler Text, a typeface designed in 1991, represents the numeral 1 as similar to a small-caps I.
1 as a resin identification code, used in recycling

1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

91 (number)

Natural number following 90 and preceding 92.

Natural number following 90 and preceding 92.

The double-struck capital N symbol, often used to denote the set of all natural numbers (see Glossary of mathematical symbols).

a triangular number.