In
arithmetic 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 ...
the seventh
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 seven instances of ''n'' together. So:
:.
Seventh powers are also formed by multiplying 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 ...
, the
square
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 a number by its
fifth power, or the
cube of a number by its
fourth power
In arithmetic and algebra, the fourth power of a number ''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. Further ...
.
The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of seventh 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 languag ...
s is:
:0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, 893871739, 1280000000, 1801088541, 2494357888, 3404825447, 4586471424, 6103515625, 8031810176, ...
In the
archaic notation of
Robert Recorde
Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557.
Biography
Born around 1512, Robert Recorde was the second and las ...
, the seventh power of a number was called the "second sursolid".
Properties
Leonard Eugene Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
studied generalizations of
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 ...
for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers (1
7 is 1, and 2
7 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. If powers of negative integers are allowed, only 12 powers are required.
The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800.
The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:
:
The two known examples of a seventh power expressible as the sum of seven seventh powers are
:
(M. Dodrill, 1999);
and
:
(Maurice Blondot, 11/14/2000);
any example with fewer terms in the sum would be a
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 ...
to
Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics 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 i ...
, which is currently only known to be false for the powers 4 and 5.
See also
*
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 ...
*
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 ...
*
Fifth power (algebra)
*
Fourth power
In arithmetic and algebra, the fourth power of a number ''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. Further ...
*
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 .
...
*
Square (algebra)
References
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers
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