Sixth power

In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube. The sequence of sixth powers of integers is: 0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304, ... They include the significant decimal numbers 106 (a million), 1006 (a short-scale trillion and long-scale billion), 10006 (a Quintillion and a long-scale trillion) and so on. The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes. In this way, they are analogous to two other classes of figurate numbers: the square triangular numbers, which are simultaneously square and triangular, and the solutions to the cannonball problem, which are simultaneously square and square-pyramidal. Because of their connection to squares and cubes, sixth powers play an important role in the study of the Mordell curves, which are elliptic curves of the form When is divisible by a sixth power, this equation can be reduced by dividing by that power to give a simpler equation of the same form. A well-known result in number theory, proven by Rudolf Fueter and Louis J. Mordell, states that, when is an integer that is not divisible by a sixth power (other than the exceptional cases and ), this equation either has no rational solutions with both and nonzero or infinitely many of them. In the archaic notation of Robert Recorde, the sixth power of a number was called the "zenzicube", meaning the square of a cube. Similarly, the notation for sixth powers used in 12th century Indian mathematics by Bhāskara II also called them either the square of a cube or the cube of a square.