Fourth power

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 biquadrates or tesseractic numbers) is:

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... (sequence A000583 in the OEIS)

PropertiesEdit

The last digit of a fourth power in decimal can only be 0 (in fact 0000), 1, 5 (in fact 0625), or 6.

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 Elkies with:

 

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:[1]

  (Allan MacLeod)
  (D.J. Bernstein)
  (D.J. Bernstein)
  (D.J. Bernstein)
  (D.J. Bernstein)
  (Roger Frye, 1988)
  (Allan MacLeod, 1998)

Equations containing a fourth powerEdit

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 radicals.

See alsoEdit

ReferencesEdit

  1. ^ Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017.