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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 ...
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows:
Mordell curve. y2 = x3 + 1, with solutions at (-1, 0), (0, 1) and (0, -1) In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer. [1] These curves were closely studied by Louis Mordell, [2] from the point of view of determining their integer points. He showed that every Mordell curve ...
Hall's conjecture. In mathematics, Hall's conjecture is an open question on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points ...
The quadratic trinomial in standard form (as from above): sum or difference of two cubes: A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable ( xn below). This form is factored as: x 2 n + r x n + s = ( x n + a 1 ) ( x n + a 2 ) , {\displaystyle x^ {2n}+rx^ {n}+s ...
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 ...
This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic. The formula can be proved as follows: Starting from the equation t 3 + pt + q = 0, let us set t = u cos θ. The idea is to choose u to make the equation coincide with the identity
Square number. Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3 .