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In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 2 3 = 8 or (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The ...
Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem (√7 is not possible due to Legendre's three-square theorem) The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4 a ( 8 b + 7 ) {\displaystyle n=4^{a ...
Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a 2 b 3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.
In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So: n 7 = n × n × n × n × n × n × n.. Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth power, or the cube of a number by its fourth power.
343 = 7 × 7 × 7 = 7 3: the cube of 7, or 7 cubed, wherein replacing two neighboring digits with their digit sums 3 + 4 and 4 + 3 yields 37: 73. Also, the product of neighboring digits 3 × 4 is 12 , like 4 × 3 , while the sum of its prime factors 7 + 7 + 7 is 21 .
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.
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The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...