Fourth power
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In arithmetic and algebra, the fourth power of a number n is the result of multiplying n by itself four times. 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 biquadratic numbers) is:
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.
