- Fourth power
In
arithmetic andalgebra , 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
integer s (also known as biquadratic numbers) is: :1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ... OEIS|id=A000583The 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)
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