fourth power

The fourth power of a number x is the number obtained multiplying x by itself three times thus: x×x×x×x. It’s more commonly denoted as x4. For example, 24=2×2×2×2=16. Since the square of a square number is a fourth power, x2x2=x2+2=x4, fourth powers are sometimes called biquadratic numbers. For example, 24=2222=42=16. The first few integer fourth powers are 1, 16, 81, 256, 625, 1296, 2401, etc., listed in A000290 of Sloane’s OEIS.

Any integer can be represented by the sum of at most 19 integer fourth powers (see Waring’s problem).

Euler’s conjecture was first disproven with fifth powers, but there are also counterexamplesMathworldPlanetmath using fourth powers. Sloane’s A003828 lists the known integers n having solutions to n4=a4+b4+c4.

The fourth power of a negative number is always a positive number; the fourth root of a negative real number is a complex numberMathworldPlanetmathPlanetmath a+bi with |a|=|b| and a0.

Title fourth power
Canonical name FourthPower
Date of creation 2013-03-22 18:25:16
Last modified on 2013-03-22 18:25:16
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 26A09
Classification msc 11A05
Synonym biquadratic number