Fourth power

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 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, ... (sequence A000583 in the OEIS)

Properties

The last digit of a fourth power in decimal can only be 0 (in fact 0000), 1, 5 (in fact 0625), or 6.

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:

${\displaystyle 20615673^{4}=18796760^{4}+15365639^{4}+2682440^{4}.}$

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:[1]

${\displaystyle 2813001^{4}=2767624^{4}+1390400^{4}+673865^{4}}$  (Allan MacLeod)
${\displaystyle 8707481^{4}=8332208^{4}+5507880^{4}+1705575^{4}}$  (D.J. Bernstein)
${\displaystyle 12197457^{4}=11289040^{4}+8282543^{4}+5870000^{4}}$  (D.J. Bernstein)
${\displaystyle 16003017^{4}=14173720^{4}+12552200^{4}+4479031^{4}}$  (D.J. Bernstein)
${\displaystyle 16430513^{4}=16281009^{4}+7028600^{4}+3642840^{4}}$  (D.J. Bernstein)
${\displaystyle 422481^{4}=414560^{4}+217519^{4}+95800^{4}}$  (Roger Frye, 1988)
${\displaystyle 638523249^{4}=630662624^{4}+275156240^{4}+219076465^{4}}$  (Allan MacLeod, 1998)

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.