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Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27. The difference between the cubes of consecutive integers can be expressed as ...
Difference of two consecutive perfect squares. The difference of two consecutive perfect squares is the sum of the two bases n and n+1. This can be seen as follows: (+) = ((+) +) ((+)) = + Therefore, the difference of two consecutive perfect squares is an odd number.
The power sum symmetric polynomial is a building block for symmetric polynomials. The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1. The Erdős–Moser equation, where m and k are positive integers, is conjectured to have no solutions other than 11 + 21 = 31. The sums of three cubes cannot ...
Fermat's last theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer cube. The first recorded proof of the exponent 3 case was given by Euler. Taxicab and Cabtaxi numbers. Taxicab numbers are numbers that can be expressed as a sum of two positive integer cubes in n distinct ways.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2) (x + 2) is a polynomial ...
Does a perfect cuboid exist? (more unsolved problems in mathematics) A perfect cuboid (also called a perfect Euler brick or perfect box) is an Euler brick whose space diagonal also has integer length. In other words, the following equation is added to the system of Diophantine equations defining an Euler brick: a 2 + b 2 + c 2 = g 2, {\displaystyle a^{2}+b^{2}+c^{2}=g^{2},} where g is the ...
Cube root. In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other ...
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. [a] Three equivalent definitions of parallelepiped are. a hexahedron with three pairs of parallel faces, a polyhedron ...