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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 ...
Hall's conjecture. In mathematics, Hall's conjecture is an open question, as of 2015, 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 ...
1729 as the sum of two positive cubes. 1729 is the smallest nontrivial taxicab number, [1] and is known as the Hardy–Ramanujan number, [2] after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: [3] [4] [5] [6]
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 ...
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.
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 ...
Fourth power. In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares. Some people refer to n4 as n “ tesseracted ”, “ hypercubed ”, “ zenzizenzic ...
Proof by exhaustion can be used to prove that if an integer is a perfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9. Proof: Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three ...