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.

The sequence of fourth powers of integers (also known as biquadratic numbers 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 two digits of a fourth power of an integer in base 10 can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only twelve possibilities:

if a number ends in 0, its fourth power ends in $00$ (in fact in $0000$ )
if a number ends in 1, 3, 7 or 9 its fourth power ends in $01$ , $21$ , $41$ , $61$ or $81$
if a number ends in 2, 4, 6, or 8 its fourth power ends in $16$ , $36$ , $56$ , $76$ or $96$
if a number ends in 5 its fourth power ends in $25$ (in fact in $0625$ )

These twelve possibilities can be conveniently expressed as 00, e1, o6 or 25 where o is an odd digit and e an even digit.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer 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 Elkiies with:

20615673^{4}=18796760^{4}+15365639^{4}+2682440^{4}.

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:^[1]

2813001^{4}=2767624^{4}+1390400^{4}+673865^{4}

(Allan MacLeod)

8707481^{4}=8332208^{4}+5507880^{4}+1705575^{4}

(D.J. Bernstein)

12197457^{4}=11289040^{4}+8282543^{4}+5870000^{4}

(D.J. Bernstein)

16003017^{4}=14173720^{4}+12552200^{4}+4479031^{4}

(D.J. Bernstein)

16430513^{4}=16281009^{4}+7028600^{4}+3642840^{4}

(D.J. Bernstein)

422481^{4}=414560^{4}+217519^{4}+95800^{4}

(Roger Frye, 1988)

638523249^{4}=630662624^{4}+275156240^{4}+219076465^{4}

(Allan MacLeod,1998)

That the equation x⁴ + y⁴ = z⁴ has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known by Fermat; see Fermat's right triangle theorem.

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

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

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular
non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect
By properties of Ω(n)	Almost prime Semiprime
By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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

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Fourth power

Properties

Equations containing a fourth power

See also

References