Fourth power

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: 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 biquadrates or tesseractic numbers) is: The last two digits of a fourth power of an integer in base 10 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: These twelve possibilities can be conveniently expressed as 00, e1, o6 or 25 where o is an odd digit and e an even digit. 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). 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 Elkies with: Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are: That the equation x4 + y4 = z4 has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known by Fermat; see Fermat's right triangle theorem.