Fourth power

This article is about mathematics. For other uses, see Fourth branch of government and Fourth Estate.

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)

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).

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven (Elkies, Frye) with:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

That the equation x⁴ + y⁴ = z⁴ has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known, 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 solvable using radicals.

References

Weisstein, Eric W. "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

This article is issued from Wikipedia - version of the 5/6/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Fourth power

Equations containing a fourth power

See also

References