sixth power

{{Short description|Result of multiplying six instances of a number}}

In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So:

:{{math|size=120%|1=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 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, ... {{OEIS|id=A001014}}

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.

Squares and cubes

The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes.{{citation|magazine=Mechanics' Magazine and Journal of Science, Arts, and Manufactures|volume=4|publisher=Knight and Lacey|date=April 30, 1825|issue=88|first=Richard|last=Dowden|page=54|url=https://books.google.com/books?id=ivs-AQAAMAAJ&pg=PA50|title=(untitled)}}

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

: $y^2=x^3+k.$

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 by Rudolf Fueter and Louis 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.{{citation

| last1 = Ireland | first1 = Kenneth F.

| last2 = Rosen | first2 = Michael I.

| isbn = 0-387-90625-8

| mr = 661047

| page = 289

| publisher = Springer-Verlag, New York-Berlin

| series = Graduate Texts in Mathematics

| title = A classical introduction to modern number theory

| url = https://books.google.com/books?id=RDzrBwAAQBAJ&pg=PA289

| volume = 84

| year = 1982}}.

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.{{citation|title=A History of Mathematical Notations|series=Dover Books on Mathematics|first=Florian|last=Cajori|author-link=Florian Cajori|publisher=Courier Corporation|year=2013|isbn=9780486161167|page=80|url=https://books.google.com/books?id=_byqAAAAQBAJ&pg=PA80}}

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.Quoted in {{cite web

| last = Meyrignac

| first = Jean-Charles

| url = http://euler.free.fr/records.htm

| title = Computing Minimal Equal Sums Of Like Powers: Best Known Solutions

| date = 14 February 2001

| access-date = 17 July 2017

}} 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.{{citation

| last1 = Vaughan | first1 = R. C.

| last2 = Wooley | first2 = T. D.

| doi = 10.1215/S0012-7094-94-07626-6

| issue = 3

| journal = Duke Mathematical Journal

| mr = 1309326

| pages = 683–710

| title = Further improvements in Waring's problem. II. Sixth powers

| volume = 76

| year = 1994}}

There are infinitely many different nontrivial solutions to the Diophantine equation{{citation

| last = Brudno | first = Simcha

| doi = 10.1090/s0025-5718-1976-0406923-6

| issue = 135

| journal = Mathematics of Computation

| mr = 0406923

| pages = 646–648

| title = Triples of sixth powers with equal sums

| volume = 30

| year = 1976| doi-access = free

}}

: $a^6+b^6+c^6=d^6+e^6+f^6.$

It has not been proven whether the equation

: $a^6+b^6=c^6+d^6$

has a nontrivial solution,{{citation

| last1 = Bremner | first1 = Andrew

| last2 = Guy | first2 = Richard K.

| doi = 10.2307/2323442

| issue = 1

| journal = American Mathematical Monthly

| mr = 1541235

| pages = 31–36

| title = Unsolved Problems: A Dozen Difficult Diophantine Dilemmas

| volume = 95

| year = 1988| jstor = 2323442

}} but the Lander, Parkin, and Selfridge conjecture would imply that it does not.

Other properties

$n^6-1$ is divisible by 7 if n isn't divisible by 7.

References

External links

{{mathworld|id=DiophantineEquation6thPowers|title=Diophantine Equation—6th Powers}}

Category:Integers

Category:Number theory

Category:Elementary arithmetic

Category:Integer sequences

Category:Unary operations

Category:Figurate numbers