Fourth power

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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:

n⁴ = n × n × n × n

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

The sequence of fourth powers of integers (also known as biquadratic numbers 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, ... (sequence A000583 in the OEIS)

Video Fourth power

Properties

The last two digits of a fourth power of an integer in base 10 can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only twelve possibilities:

if a number ends in 0, its fourth power ends in $00$ (in fact in $0000$ )
if a number ends in 1, 3, 7 or 9 its fourth power ends in $01$ , $21$ , $41$ , $61$ or $81$
if a number ends in 2, 4, 6, or 8 its fourth power ends in $16$ , $36$ , $56$ , $76$ or $96$
if a number ends in 5 its fourth power ends in $25$ (in fact in $0625$ )

These twelve possibilities can be conveniently expressed as 00, e1, o6 or 25 where o is an odd digit and e an even digit.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer 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 Elkiies with:

20615673^{4}=18796760^{4}+15365639^{4}+2682440^{4}.

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:

2813001^{4}=2767624^{4}+1390400^{4}+673865^{4}

(Allan MacLeod)

8707481^{4}=8332208^{4}+5507880^{4}+1705575^{4}

(D.J. Bernstein)

12197457^{4}=11289040^{4}+8282543^{4}+5870000^{4}

(D.J. Bernstein)

16003017^{4}=14173720^{4}+12552200^{4}+4479031^{4}

(D.J. Bernstein)

16430513^{4}=16281009^{4}+7028600^{4}+3642840^{4}

(D.J. Bernstein)

422481^{4}=414560^{4}+217519^{4}+95800^{4}

(Roger Frye, 1988)

638523249^{4}=630662624^{4}+275156240^{4}+219076465^{4}

(Allan MacLeod,1998)

That the equation x⁴ + y⁴ = z⁴ has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known by Fermat; see Fermat's right triangle theorem.

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Equations containing a fourth power

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.