In
arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
...
and
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, the fifth power or sursolid of a
number
A number is a mathematical object used to count, measure, and label. The most basic 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 can ...
''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
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of a number by its
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
.
The sequence of fifth powers of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is:
:0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, ...
Properties
For any integer ''n'', the last
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 (''decimal fractions'') of th ...
digit of ''n''
5 is the same as the last (decimal) digit of ''n'', i.e.
:
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 t ...
, there is no general
algebraic formula
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations:
addition (+), subtraction (-), multiplication (×), division (÷), whole number po ...
(formula expressed in terms of
radical expressions) for the solution of
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
s containing a fifth power of the
unknown
Unknown or The Unknown may refer to:
Film and television Film
* The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), Australian silent film
* The Unknown (1915 drama film), ''The Unknown'' (1915 drama film), American silent drama ...
as their highest power. This is the lowest power for which this is true. See
quintic equation
In mathematics, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other word ...
,
sextic equation
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precise ...
, and
septic equation
In algebra, a septic equation is an equation of the form
:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,
where .
A septic function is a function of the form
:f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\,
where . In other words, it is a polynomial of ...
.
Along with the fourth power, the fifth power is one of two powers ''k'' that can be expressed as the sum of ''k'' − 1 other ''k''-th powers, providing counterexamples to
Euler's sum of powers conjecture
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers ...
. Specifically,
: (Lander & Parkin, 1966)
See also
*
Eighth power
*
Seventh power
*
Sixth power
*
Fourth power
*
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third exponentiation, power, that is, the result of multiplying three instances of together.
The cube of a number is denoted , using a superscript 3, for example . The cube Mathematical ...
*
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square ...
*
Perfect power
Footnotes
References
*
*
*
*
*
*
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers
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