Fourth power

For other uses, see Fourth power (disambiguation).

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, 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, 1, 5, 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

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

1. 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 (2^kn)⁴ = 16^kn⁴ 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.
2. Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions" . Retrieved 17 July 2017.

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