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:
Properties
editThe 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.[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:[2]
- 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
editFourth-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
editReferences
edit- ^ An odd fourth power is the square of an odd square number. All odd squares are congruent to 1 modulo 8, and (8n+1)2 = 64n2 + 16n + 1 = 16(4n2 + 1) + 1, meaning that all fourth powers are congruent to 1 modulo 16. Even fourth powers (excluding zero) are equal to (2kn)4 = 16kn4 for some positive integer k and odd integer n, meaning that an even fourth power can be represented as an odd fourth power multiplied by a power of 16.
- ^ Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017.