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 two digits of a fourth power of an integer in senary or decimal can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only eight possibilities in senary, and only twelve possibilities in decimal.

In senary
• if a number ends in 0, its fourth power ends in ${\displaystyle 00}$  (in fact in ${\displaystyle 0000}$ )
• if a number ends in 1 or 5, its fourth power ends in ${\displaystyle 01}$ , ${\displaystyle 21}$  or ${\displaystyle 41}$
• if a number ends in 2 or 4, its fourth power ends in ${\displaystyle 04}$ , ${\displaystyle 24}$  or ${\displaystyle 44}$
• if a number ends in 3, its fourth power ends in ${\displaystyle 13}$  (in fact in ${\displaystyle 0213}$ )
In decimal
• if a number ends in 0, its fourth power ends in ${\displaystyle 00}$  (in fact in ${\displaystyle 0000}$ )
• if a number ends in 1, 3, 7 or 9 its fourth power ends in ${\displaystyle 01}$ , ${\displaystyle 21}$ , ${\displaystyle 41}$ , ${\displaystyle 61}$  or ${\displaystyle 81}$
• if a number ends in 2, 4, 6, or 8 its fourth power ends in ${\displaystyle 16}$ , ${\displaystyle 36}$ , ${\displaystyle 56}$ , ${\displaystyle 76}$  or ${\displaystyle 96}$
• if a number ends in 5 its fourth power ends in ${\displaystyle 25}$  (in fact in ${\displaystyle 0625}$ )
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 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.