Fourth power


In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:
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 is:

Properties

The last two digits of a fourth power of an integer in senary or decimal can be easily shown to be restricted to only eight possibilities in senary, and only twelve possibilities in decimal.
;In senary
;In decimal
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.
Fermat knew that a fourth power cannot be the sum of two other fourth powers. 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:
Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:

Equations containing a fourth power

s, which contain a fourth degree polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using radicals.