In
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the fourth
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
of a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''
4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
. Furthermore, they are
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of squares.
The sequence of fourth powers of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s (also known as biquadrates or
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
ic 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, ... .
Properties
The last digit of a fourth power in
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
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
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...
).
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
knew that a fourth power cannot be the sum of two other fourth powers (the
''n'' = 4 case of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
; see
Fermat's right triangle theorem
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one o ...
).
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
conjectured
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 199 ...
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
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
s for exponent four, some of which are:
[Quoted in
]
: (Allan MacLeod)
: (D.J. Bernstein)
: (D.J. Bernstein)
: (D.J. Bernstein)
: (D.J. Bernstein)
: (Roger Frye, 1988)
: (Allan MacLeod, 1998)
Equations containing a fourth power
Fourth-degree equations, which contain a fourth
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
(but no higher)
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
are, by the
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
, the highest degree equations having a general solution using
radicals.
See also
*
Square (algebra)
In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as exponentiation, raising to the power 2 (number), 2, and is denoted by a ...
*
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or .
...
*
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
*
Fifth power (algebra)
In arithmetic and algebra, the fifth power or sursolid of a number ''n'' is the result of multiplying five instances of ''n'' together:
:.
Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its ...
*
Sixth power In arithmetic and algebra the sixth power of a number ''n'' is the result of multiplying six instances of ''n'' together. So:
:.
Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourt ...
*
Seventh power
In arithmetic and algebra the seventh power of a number ''n'' is the result of multiplying seven instances of ''n'' together. So:
:.
Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth p ...
*
Eighth power
In arithmetic and algebra the eighth power of a number ''n'' is the result of multiplying eight instances of ''n'' together. So:
:.
Eighth powers are also formed by multiplying a number by its seventh power, or the fourth power of a number by it ...
*
Perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...
References
*
{{Classes of natural numbers
Figurate numbers
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations