- n4 = n × n × n × n
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.
Equations to the fourth power
Equations with a fourth degree polynomial on a side of the equation are proven to be the highest degree polynomial solvable using radicals. Another way is to find two factors of the polynomial and divide the original polynomial by them. This results in a quadratic equation which can be easily solved by either the quadratic formula, by completing the square, by using radicals or by factoring back into two binomials.
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