Sixth power
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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 sixth
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 six instances of ''n'' together. So: :. Sixth powers can be formed by multiplying a number by its fifth power, multiplying 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 adj ...
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 ...
, by cubing a square, or by squaring a cube. The sequence of sixth 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 is: :0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304, ... They include the significant
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 ...
numbers 106 (a
million One million (1,000,000), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian ''millione'' (''milione'' in modern Italian), from ''mille'', "thousand", plus the au ...
), 1006 (a short-scale trillion and long-scale billion), 10006 (a long-scale trillion) and so on.


Squares and cubes

The sixth powers of integers can be characterized as the numbers that are simultaneously squares and cubes. In this way, they are analogous to two other classes of
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
s: the
square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...
s, which are simultaneously square and triangular, and the solutions to the
cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
, which are simultaneously square and square-pyramidal. Because of their connection to squares and cubes, sixth powers play an important role in the study of the
Mordell curve In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( ...
s, which are
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s of the form :y^2=x^3+k. When k is divisible by a sixth power, this equation can be reduced by dividing by that power to give a simpler equation of the same form. A well-known result in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
,
proven Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants. The church and parish of Proven are named after Saint Victor. The Saint Victor Chur ...
by
Rudolf Fueter Karl Rudolf Fueter (30 June 1880 – 9 August 1950) was a Swiss mathematician, known for his work on number theory. Biography After a year of graduate study of mathematics in Basel, Fueter began study in 1899 at the University of Göttingen and ...
and Louis J. Mordell, states that, when k is an integer that is not divisible by a sixth power (other than the exceptional cases k=1 and k=-432), this equation either has no
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
solutions with both x and y nonzero or infinitely many of them. 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 and minus signs, plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde w ...
, the sixth power of a number was called the "zenzicube", meaning the square of a cube. Similarly, the notation for sixth powers used in 12th century
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
by
Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...
also called them either the square of a cube or the cube of a square.


Sums

There are numerous known examples of sixth powers that can be expressed as the sum of seven other sixth powers, but no examples are yet known of a sixth power expressible as the sum of just six sixth powers.Quoted in This makes it unique among the powers with exponent ''k'' = 1, 2, ... , 8, the others of which can each be expressed as the sum of ''k'' other ''k''-th powers, and some of which (in violation of
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 is ...
) can be expressed as a sum of even fewer ''k''-th powers. In connection with
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 ...
, every sufficiently large integer can be represented as a sum of at most 24 sixth powers of integers. There are infinitely many different nontrivial solutions to the
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
:a^6+b^6+c^6=d^6+e^6+f^6. It has not been proven whether the equation :a^6+b^6=c^6+d^6 has a nontrivial solution, but the
Lander, Parkin, and Selfridge conjecture The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some ' ...
would imply that it does not.


Other properties

* n^6-1 is divisible by 7 iff n isn't divisible by 7.


See also

*
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 precis ...
*
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 ...
*
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 ...
*
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 ...
*
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) 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 ...


References


External links

* {{Classes of natural numbers Integers Number theory Elementary arithmetic Integer sequences Unary operations Figurate numbers