• Source: 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.
      Some people refer to n4 as n “tesseracted”, “hypercubed”, “zenzizenzic”, “biquadrate” or “supercubed” instead of “to the power of 4”.
      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).


      Properties


      The last digit of a fourth power in decimal can only be 0 (in fact 0000), 1, 5 (in fact 0625), or 6.
      In hexadecimal the last nonzero digit of a fourth power is always 1.
      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:

      206156734 = 187967604 + 153656394 + 26824404.
      Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:

      28130014 = 27676244 + 13904004 + 6738654 (Allan MacLeod)
      87074814 = 83322084 + 55078804 + 17055754 (D.J. Bernstein)
      121974574 = 112890404 + 82825434 + 58700004 (D.J. Bernstein)
      160030174 = 141737204 + 125522004 + 44790314 (D.J. Bernstein)
      164305134 = 162810094 + 70286004 + 36428404 (D.J. Bernstein)
      4224814 = 4145604 + 2175194 + 958004 (Roger Frye, 1988)
      6385232494 = 6306626244 + 2751562404 + 2190764654 (Allan MacLeod, 1998)


      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 having a general solution using radicals.


      See also


      Square (algebra)
      Cube (algebra)
      Exponentiation
      Fifth power (algebra)
      Sixth power
      Seventh power
      Eighth power
      Perfect power


      References



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

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