Editing Fourth power

{{Short description|Result of multiplying four instances of a number together}}
{{other uses}}
In [[arithmetic]] and [[algebra]], the '''fourth power''' of a [[number]] ''n'' is the result of multiplying four instances of ''n'' together. So:

:''n''<sup>4</sup> = ''n'' × ''n'' × ''n'' × ''n''

Fourth [[exponentiation|powers]] are also formed by multiplying a number by its [[cube (algebra)|cube]]. Furthermore, they are [[square number|squares]] of squares.

Some people refer to ''n''<sup>4</sup> as n ''[[Tesseract|tesseracted]]'', ''[[Hypercube|hypercubed]]'', ''[[Zenzizenzizenzic#History|zenzizenzic]]'', ''[[Biquadratic|biquadrate]]'' or ''supercubed'' instead of “to the power of 4”. 

The sequence of fourth powers of [[integer]]s, known as '''biquadrates''' or '''tesseractic numbers''', is:
 
:0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... {{OEIS|id=A000583}}.

==Properties==
The last digit of a fourth power in [[decimal]] can only be 0, 1, 5, or 6.

In [[hexadecimal]] the last nonzero digit of a fourth power is always 1.<ref>An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)<sup>2</sup> = 64n<sup>2</sup> + 16n + 1 = 16(4n<sup>2</sup> + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2<sup>k</sup>n)<sup>4</sup> = 16<sup>k</sup>n<sup>4</sup> for some positive integer k and odd integer n, meaning that an even fourth power can be represented as an odd fourth power multiplied by a power of 16.</ref>

Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see [[Waring's problem]]).

[[Pierre de Fermat|Fermat]] knew that a fourth power cannot be the sum of two other fourth powers (the [[Proof of Fermat's Last Theorem for specific exponents#n = 4|''n'' = 4 case]] of [[Fermat's Last Theorem]]; see [[Fermat's right triangle theorem]]). [[Euler]] [[Euler's sum of powers conjecture|conjectured]] that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986,  this was disproven by [[Noam Elkies|Elkies]] with:

: {{math|1=20615673<sup>4</sup> =  18796760<sup>4</sup> + 15365639<sup>4</sup> + 2682440<sup>4</sup>}}.

Elkies showed that there are infinitely many other [[counterexample]]s for exponent four, some of which are:<ref name=meyrignac>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
}}</ref>

:{{math|1=2813001<sup>4</sup> = 2767624<sup>4</sup> + 1390400<sup>4</sup> + 673865<sup>4</sup>}} (Allan MacLeod)
:{{math|1= 8707481<sup>4</sup> = 8332208<sup>4</sup> + 5507880<sup>4</sup> + 1705575<sup>4</sup>}} (D.J. Bernstein)
:{{math|1=12197457<sup>4</sup> = 11289040<sup>4</sup> + 8282543<sup>4</sup> + 5870000<sup>4</sup>}} (D.J. Bernstein)
:{{math|1=16003017<sup>4</sup> = 14173720<sup>4</sup> + 12552200<sup>4</sup> + 4479031<sup>4</sup>}} (D.J. Bernstein)
:{{math|1=16430513<sup>4</sup> = 16281009<sup>4</sup> + 7028600<sup>4</sup> + 3642840<sup>4</sup>}} (D.J. Bernstein)
:{{math|1= 422481<sup>4</sup> = 414560<sup>4</sup> + 217519<sup>4</sup> + 95800<sup>4</sup>}} (Roger Frye, 1988)
:{{math|1= 638523249<sup>4</sup> = 630662624<sup>4</sup> + 275156240<sup>4</sup> + 219076465<sup>4</sup>}} (Allan MacLeod, 1998)

==Equations containing a fourth power==
[[Fourth-degree equation]]s, which contain a fourth [[degree of a polynomial|degree]] (but no higher) [[polynomial]] are, by the [[Abel–Ruffini theorem]], the highest degree equations having a general solution using [[Nth root|radicals]].

== See also ==
*[[Square (algebra)]]
*[[Cube (algebra)]]
*[[Exponentiation]]
*[[Fifth power (algebra)]]
*[[Sixth power]]
*[[Seventh power]]
*[[Eighth power]]
*[[Perfect power]]

==References==
<references />
*{{MathWorld|title=Biquadratic Number|urlname=BiquadraticNumber}}

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Figurate numbers]]
[[Category:Integers]]
[[Category:Number theory]]
[[Category:Elementary arithmetic]]
[[Category:Integer sequences]]
[[Category:Unary operations]]