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 power, or the cube of a number by its fourth power.
The sequence of seventh powers of integers is:
In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid". [1]
Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers [2] (17 is 1, and 27 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. [3] If powers of negative integers are allowed, only 12 powers are required. [4]
The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. [5]
The smallest seventh power that can be represented as a sum of eight distinct seventh powers is: [6]
The two known examples of a seventh power expressible as the sum of seven seventh powers are
and
any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.
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 power, or the cube of a number by its fourth power.
The sequence of seventh powers of integers is:
In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid". [1]
Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers [2] (17 is 1, and 27 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. [3] If powers of negative integers are allowed, only 12 powers are required. [4]
The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. [5]
The smallest seventh power that can be represented as a sum of eight distinct seventh powers is: [6]
The two known examples of a seventh power expressible as the sum of seven seventh powers are
and
any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.