Inarithmetic andalgebra thesixth power of anumbern is the result of multiplying six instances ofn together. So:

n⁶ =n ×n ×n ×n ×n ×n.

Sixthpowers can be formed by multiplying a number by itsfifth power, multiplying thesquare of a number by itsfourth power, bycubing a square, or by squaring a cube.

The sequence of sixth powers ofintegers are:

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, ... (sequenceA001014 in theOEIS)

They include the significantdecimal numbers 10⁶ (amillion), 100⁶ (ashort-scale trillion and long-scale billion), 1000⁶ (aquintillion and along-scale trillion) and so on.

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 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 curves of the form

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

When${\displaystyle k}$ 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 innumber theory,proven byRudolf Fueter andLouis J. Mordell, states that, when${\displaystyle k}$ is an integer that is not divisible by a sixth power (other than the exceptional cases${\displaystyle k=1}$ and${\displaystyle k=-432}$), this equation either has norational solutions with both${\displaystyle x}$ and${\displaystyle y}$ nonzero or infinitely many of them.^[2]

In thearchaic notation ofRobert 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 mathematics byBhāskara II also called them either the square of a cube or the cube of a square.^[3]

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 exponentk = 1, 2, ... , 8, the others of which can each be expressed as the sum ofk otherk-th powers, and some of which (in violation ofEuler's sum of powers conjecture) can be expressed as a sum of even fewerk-th powers.

In connection withWaring'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 theDiophantine equation^[6]

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

It has not been proven whether the equation

${\displaystyle a^6+b^6=c^6+d^6}$

has a nontrivial solution,^[7] but theLander, Parkin, and Selfridge conjecture would imply that it does not.

• ${\displaystyle n^6-1}$ is divisible by 7 ifn isn't divisible by 7.

• Sextic equation
• Eighth power
• Seventh power
• Fifth power (algebra)
• Fourth power
• Cube (algebra)
• Square (algebra)

• Weisstein, Eric W."Diophantine Equation—6th Powers".MathWorld.

• Integers
• Number theory
• Elementary arithmetic
• Integer sequences
• Unary operations
• Figurate numbers

• Articles with short description
• Short description matches Wikidata