Doubling the cube given length times . In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is three repetitions of the same interval. In this sense, the problem of doubling the cube is solved by the major third in equal temperament. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by , the side length of the Delian cube. Doubling the cube Doubling the cube, also known as the Delian problem, is
Root of unity modulo n it is not a primitive root, then there would be some divisor ℓ of "k", with formula_57. In order to exclude this possibility, one has only to check for a few ℓ's equal "k" divided by a prime. That is, what needs to be checked is: Among the primitive "k"-th roots of unity, the primitive formula_23-th roots are most frequent. It is thus recommended to try some integers for being a primitive formula_23-th root, what will succeed quickly. For a primitive formula_23-th root "x", the number formula_30 is a primitive formula_25-th root of unity. If "k" does not divide formula_23,