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What is the power of 6 called?

Squares and cubes 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.

en.wikipedia.org - Sixth power - Wikipedia

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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:

They include the significant decimal numbers 106 (a million), 1006 (a short-scale trillion and long-scale billion), 10006 (a long-scale trillion) and so on.

Squares and cubes [ edit ]

The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes.[1] 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

y 2 = x 3 + k . {\displaystyle y^{2}=x^{3}+k.}

When k {\displaystyle k} 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 k {\displaystyle k} is an integer that is not divisible by a sixth power (other than the exceptional cases k = 1 {\displaystyle k=1} and k = − 432 {\displaystyle k=-432} ), this equation either has no rational solutions with both x {\displaystyle x} and y {\displaystyle y} nonzero or infinitely many of them.[2] 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.[3]

Sums [ edit ]

There are numerous known examples of sixth powers that can be expressed as the sum of seven other sixth powers, but no examples are yet known of a sixth power expressible as the sum of just six sixth powers.[4] This makes it unique among the powers with exponent k = 1, 2, ... , 8, the others of which can each be expressed as the sum of k other k-th powers, and some of which (in violation of Euler's sum of powers conjecture) can be expressed as a sum of even fewer k-th powers. In connection with Waring's problem, every sufficiently large integer can be represented as a sum of at most 24 sixth powers of integers.[5]

There are infinitely many different nontrivial solutions to the Diophantine equation[6]

a 6 + b 6 + c 6 = d 6 + e 6 + f 6 . {\displaystyle a^{6}+b^{6}+c^{6}=d^{6}+e^{6}+f^{6}.}

It has not been proven whether the equation

a 6 + b 6 = c 6 + d 6 {\displaystyle a^{6}+b^{6}=c^{6}+d^{6}} has a nontrivial solution,[7] but the Lander, Parkin, and Selfridge conjecture would imply that it does not.