Sixth power

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64 (2 ) and 729 (3 ) cubelets arranged as cubes (2 and 3 , respectively) and as squares (2 and 3 , respectively) Sixth power example.svg
64 (2 ) and 729 (3 ) cubelets arranged as cubes (2 and 3 , respectively) and as squares (2 and 3 , respectively)

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

Contents

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, ... (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.

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

When 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 is an integer that is not divisible by a sixth power (other than the exceptional cases and ), this equation either has no rational solutions with both and 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]

It has not been proven whether the equation

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

Other properties

See also

Related Research Articles

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In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:

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In number theory, the Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.

In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together:

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

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In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer to equal such a sum is that cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient.

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

References

  1. ↑ Dowden, Richard (April 30, 1825), "(untitled)", Mechanics' Magazine and Journal of Science, Arts, and Manufactures, vol. 4, no. 88, Knight and Lacey, p. 54
  2. ↑ Ireland, Kenneth F.; Rosen, Michael I. (1982), A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, p. 289, ISBN   0-387-90625-8, MR   0661047 .
  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" . Retrieved 17 July 2017.
  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, MR   1309326
  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 , MR   0406923
  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, JSTOR   2323442, MR   1541235
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