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The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
Alternatively, one can decompose the table into a sequence of nested gnomons, each consisting of the products in which the larger of the two terms is some fixed value. The sum within each gmonon is a cube, so the sum of the whole table is a sum of cubes. [6] Visual demonstration that the square of a triangular number equals a sum of cubes.
A Cabtaxi number is the smallest positive number that can be expressed as a sum of two integer cubes in n ways, allowing the cubes to be negative or zero as well as positive. The smallest cabtaxi number after Cabtaxi(1) = 0, is Cabtaxi(2) = 91, [ 5 ] expressed as:
The formula for a general cuban prime of this kind can be simplified to + +. This is exactly the general form of a centered hexagonal number ; that is, all of these cuban primes are centered hexagonal.
Because of the factorization (2n + 1)(n 2 + n + 1), it is impossible for a centered cube number to be a prime number. [3] The only centered cube numbers which are also the square numbers are 1 and 9, [4] [5] which can be shown by solving x 2 = y 3 + 3y, the only integer solutions being (x,y) from {(0,0), (1,2), (3,6), (12,42)}, By substituting a=(x-1)/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1.
The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of C n for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).
According to an anecdote of uncertain reliability, [1] in primary school Carl Friedrich Gauss reinvented the formula (+) for summing the integers from 1 through , for the case =, by grouping the numbers from both ends of the sequence into pairs summing to 101 and multiplying by the number of pairs.
Semi-log plot of solutions of + + = for integer , , and , and .Green bands denote values of proven not to have a solution.. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.
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