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 {{mvar|n}} is the result of multiplying four instances of {{mvar|n}} together: {{math|1=''n''⁴ = ''n'' × ''n'' × ''n'' × ''n''}}.

Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.

Some people refer to {{math|''n''⁴}} as n ''tesseracted'', ''hypercubed'', ''zenzizenzic'', ''biquadrate'' or ''supercubed'' instead of "to the power of 4".

The sequence of fourth powers of integers, 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)² = 64n² + 16n + 1 = 16(4n² + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2^kn)⁴ = 16^kn⁴ 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).

Fermat knew that a fourth power cannot be the sum of two other fourth powers (the ''n'' = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler 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 Elkies with:

: {{math|1=20615673⁴ = 18796760⁴ + 15365639⁴ + 2682440⁴}}.

Elkies showed that there are infinitely many other counterexamples 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=2813001⁴ = 2767624⁴ + 1390400⁴ + 673865⁴}} (Allan MacLeod) :{{math|1= 8707481⁴ = 8332208⁴ + 5507880⁴ + 1705575⁴}} (D.J. Bernstein) :{{math|1=12197457⁴ = 11289040⁴ + 8282543⁴ + 5870000⁴}} (D.J. Bernstein) :{{math|1=16003017⁴ = 14173720⁴ + 12552200⁴ + 4479031⁴}} (D.J. Bernstein) :{{math|1=16430513⁴ = 16281009⁴ + 7028600⁴ + 3642840⁴}} (D.J. Bernstein) :{{math|1= 422481⁴ = 414560⁴ + 217519⁴ + 95800⁴}} (Roger Frye, 1988) :{{math|1= 638523249⁴ = 630662624⁴ + 275156240⁴ + 219076465⁴}} (Allan MacLeod, 1998)

Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using 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