Search results
Results from the WOW.Com Content Network
In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers; this is equivalent to the fact that 8n + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem. [4]
Consequently, a square number is also triangular if and only if + is square, that is, there are numbers and such that =. This is an instance of the Pell equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} .
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers : if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum). A000292: Square pyramidal numbers: 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... n(n + 1)(2n + 1) / 6 : The number of stacked spheres in a pyramid with a square base. A000330: Cube numbers n 3: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ... n 3 = n × n × n ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The space diagonal of the unit cube is √ 3. Distances between vertices of a double unit cube are square roots of the first six natural numbers, including the square root of 3 (√7 is not possible due to Legendre's three-square theorem) This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio √ 3.