The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes. In this way, they are related to two other classes offigurate numbers: thesquare triangular numbers, which are simultaneously square and triangular, and the solutions to thecannonball 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 theMordell curves, which areelliptic curvesof the form
y2=x3+k.
When
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 byRudolf FueterandLouis J. Mordell, states that, when
k
is an integer that is not divisible by a sixth power (other than the exceptional cases
k=1
and
k=−432
), this equation either has no rational solutions with both
x
and
y
nonzero or infinitely many of them.
In thearchaic notationofRobert 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 centuryIndian mathematicsbyBhāskara IIalso called them either the square of a cube or the cube of a square.
