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In arithmetic and algebra, the fourth power of a number is the result of multiplying four instances of together: .
Fourth exponentiation are also formed by multiplying a number by its cube. Furthermore, they are square number of squares.
Some people refer to as n Tesseract, Hypercube, zenzizenzic, Biquadratic or supercubed instead of "to the power of 4".
The sequence of fourth powers of , known as biquadrates or tesseractic numbers, is:
In hexadecimal the last nonzero digit of a fourth power is always 1.An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)2 = 64n2 + 16n + 1 = 16(4n2 + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2kn)4 = 16kn4 for some positive integer k and odd integer n, meaning that an even fourth power can be represented as an odd fourth power multiplied by a power of 16.
Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as the sum of at most 16 fourth powers (see Waring's problem).
Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). 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 Noam Elkies with:
Elkies showed that there are infinitely many other for exponent four, some of which are:Quoted in
Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using Nth root.
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