Fourth power

This article is about mathematics. For other uses, see Fourth branch and Fourth Estate.

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. 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 or tesseractic numbers) is:

1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, (sequence A000583 in OEIS)

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:

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 (Elkies, Frye) with:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

That the equation x⁴ + y⁴ = z⁴ has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known, see Fermat's right triangle theorem.

Equations containing a fourth power

Fourth degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel-Ruffini theorem, the highest degree equations solvable using radicals.

See also

• Square (algebra)
• Cube (algebra)
• Exponentiation
• Fifth power (algebra)
• Perfect power

References

• Weisstein, Eric W., "Biquadratic Number", MathWorld.