Английская Википедия:Fourth power
Шаблон:Short description Шаблон:About 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, ... Шаблон: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.
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]
- Шаблон:Math (Allan MacLeod)
- Шаблон:Math (D.J. Bernstein)
- Шаблон:Math (D.J. Bernstein)
- Шаблон:Math (D.J. Bernstein)
- Шаблон:Math (D.J. Bernstein)
- Шаблон:Math (Roger Frye, 1988)
- Шаблон:Math (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
- ↑ Quoted in Шаблон:Cite web
Шаблон:Figurate numbers Шаблон:Classes of natural numbers
