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.^[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 = 18796760 415365639426824404

.

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:^[2]

28130014 = 2767624 4139040046738654

(Allan MacLeod)

87074814 = 8332208 45507880417055754

(D.J. Bernstein)

121974574 = 11289040 48282543458700004

(D.J. Bernstein)

160030174 = 14173720 412552200444790314

(D.J. Bernstein)

164305134 = 16281009 47028600436428404

(D.J. Bernstein)

4224814 = 414560 42175194958004

(Roger Frye, 1988)

6385232494 = 630662624 427515624042190764654

(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.

References

^ 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.
^ 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.

[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.

[meyrignac-2] Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017.

[1]

[2]

Properties

Equations containing a fourth power

See also

References