Sixth power

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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:

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 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.[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]


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


  1. Dowden, Richard (April 30, 1825), "(untitled)", Mechanics' Magazine and Journal of Science, Arts, and Manufactures (Knight and Lacey) 4 (88): 54, 
  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, .
  3. Cajori, Florian (2013), A History of Mathematical Notations, Dover Books on Mathematics, Courier Corporation, p. 80, ISBN 9780486161167, 
  4. Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". 
  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