Sixth power

64 (26) and 729 (36) cubelets arranged as cubes (223 and 323, respectively) and as squares (232 and 332, respectively)

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, ... (sequence A001014 in the OEIS)

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.

## Contents

* 1 Squares and cubes
* 2 Sums
* 3 Other properties
* 4 See also
* 5 References
* 6 External links

## Squares and cubes

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

[math]\displaystyle{ y^2=x^3+k. }[/math]

When [math]\displaystyle{ k }[/math] 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 [math]\displaystyle{ k }[/math] is an integer that is not divisible by a sixth power (other than the exceptional cases [math]\displaystyle{ k=1 }[/math] and [math]\displaystyle{ k=-432 }[/math]), this equation either has no rational solutions with both [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] 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

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]

[math]\displaystyle{ a^6+b^6+c^6=d^6+e^6+f^6. }[/math]

It has not been proven whether the equation

[math]\displaystyle{ a^6+b^6=c^6+d^6 }[/math]

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

## Other properties

* [math]\displaystyle{ n^6-1 }[/math] is divisible by 7 iff n isn't divisible by 7.

## See also

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

## References

1. ↑ Dowden, Richard (April 30, 1825), "(untitled)", Mechanics' Magazine and Journal of Science, Arts, and Manufactures (Knight and Lacey) 4 (88): 54, https://books.google.com/books?id=ivs-AQAAMAAJ&pg=PA50
2. ↑ Ireland, Kenneth F.; Rosen, Michael I. (1982), A classical introduction to modern number theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York-Berlin, p. 289, ISBN 0-387-90625-8, https://books.google.com/books?id=RDzrBwAAQBAJ&pg=PA289 .
3. ↑ Cajori, Florian (2013), A History of Mathematical Notations, Dover Books on Mathematics, Courier Corporation, p. 80, ISBN 9780486161167, https://books.google.com/books?id=_byqAAAAQBAJ&pg=PA80
4. ↑ Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". http://euler.free.fr/records.htm.
5. ↑ Vaughan, R. C.; Wooley, T. D. (1994), "Further improvements in Waring's problem. II. Sixth powers", Duke Mathematical Journal 76 (3): 683–710, doi:10.1215/S0012-7094-94-07626-6
6. ↑ Brudno, Simcha (1976), "Triples of sixth powers with equal sums", Mathematics of Computation 30 (135): 646–648, doi:10.1090/s0025-5718-1976-0406923-6
7. ↑ Bremner, Andrew; Guy, Richard K. (1988), "Unsolved Problems: A Dozen Difficult Diophantine Dilemmas", American Mathematical Monthly 95 (1): 31–36, doi:10.2307/2323442

## External links

* Weisstein, Eric W.. "Diophantine Equation—6th Powers". http://mathworld.wolfram.com/DiophantineEquation6thPowers.html.

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Classes of natural numbers

Powers and
related numbers|

* Achilles
* Power of 2
* Power of 10
* Square
* Cube
* Fourth power
* Fifth power
* Sixth power
* Seventh power
* Perfect power
* Powerful
* Prime power

Of the form
a × 2b ± 1|

* Cullen
* Double Mersenne
* Fermat
* Mersenne
* Proth
* Thabit
* Woodall

Other polynomial
numbers|

* Carol
* Hilbert
* Idoneal
* Kynea
* Leyland
* Loeschian
* Lucky numbers of Euler

Recursively defined
numbers|

* Fibonacci
* Jacobsthal
* Leonardo
* Lucas
* Padovan
* Pell
* Perrin

Possessing a
specific set
of other numbers|

* Knödel
* Riesel
* Sierpinski

Expressible via
specific sums|

* Nonhypotenuse
* Polite
* Practical
* Primary pseudoperfect
* Ulam
* Wolstenholme

Figurate numbers|

| 2-dimensional|

| centered|

* Centered triangular
* Centered square
* Centered pentagonal
* Centered hexagonal
* Centered heptagonal
* Centered octagonal
* Centered nonagonal
* Centered decagonal
* Star

|

non-centered|

* Triangular
* Square
* Square triangular
* Pentagonal
* Hexagonal
* Heptagonal
* Octagonal
* Nonagonal
* Decagonal
* Dodecagonal

3-dimensional|

| centered|

* Centered tetrahedral
* Centered cube
* Centered octahedral
* Centered dodecahedral
* Centered icosahedral

|

non-centered|

* Tetrahedral
* Cube
* Octahedral
* Dodecahedral
* Icosahedral
* Stella octangula

pyramidal|

* Square pyramidal
* Pentagonal pyramidal
* Hexagonal pyramidal
* Heptagonal pyramidal

4-dimensional|

| centered|

* Centered pentachoric
* Squared triangular

|

non-centered|

* Pentatope
* 4-cube

Base-dependent
numbers|

| Arithmetic functions and dynamics|

* Digit sum
* Digital root
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* Persistence
* Additive
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|

P-adic numbers related|

* Automorphic

Divisor related|

* Equidigital
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* Smith

Digit-composition related|

* Palindromic
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Digit-permutation related|

* Cyclic
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Other|

* Friedman
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Combinatorial
numbers|

* Bell
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Pseudoprimes|

* Carmichael number
* Catalan pseudoprime
* Elliptic pseudoprime
* Euler pseudoprime
* Euler–Jacobi pseudoprime
* Fermat pseudoprime
* Frobenius pseudoprime
* Lucas pseudoprime
* Somer–Lucas pseudoprime
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Arithmetic functions|

| By properties of σ(n)|

* Abundant
* Arithmetic
* Colossally abundant
* Descartes
* Hemiperfect
* Highly abundant
* Highly composite
* Hyperperfect
* Multiply perfect
* Perfect
* Practical number
* Primitive abundant
* Quasiperfect
* Refactorable
* Sublime
* Superabundant
* Superior highly composite
* Superperfect
* Untouchable

|

By properties of Ω(n)|

* Almost prime
* Semiprime

By properties of φ(n)|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

By properties of s(n)|

* Almost perfect
* Amicable
* Betrothed
* Deficient
* Perfect
* Semiperfect
* Sociable

Other|

* Euclid
* Fortunate

Dividing a quotient|

* Wieferich
* Wall–Sun–Sun
* Wolstenholme prime
* Wilson

Other prime factor
or divisor related
numbers|

* Blum
* Erdős–Nicolas
* Erdős–Woods
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* Sociable
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Binary numbers|

* Evil
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Generated
via a sieve|

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Sorting related|

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Natural language related|

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