Fourth power

In arithmetic and algebra, the fourth power of a number "n" is the result of multiplying "n" by itself four times. So:

:"n"⁴ = "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) is: :1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ... OEIS|id=A000583

The last two digits of a fourth power of an integer 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:

:00, 01, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96

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).

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven with:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

See also

*square number
*cube (arithmetic)

References

*MathWorld|title=Biquadratic Number|urlname=BiquadraticNumber

External links

* [http://www.ascz85.dsl.pipex.com/files/4th-powers.txt 1 million fourth powers] (36.9 MBs)