What is the power of 7 called?

(sequence A001015 in the OEIS) In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid".

en.wikipedia.org - Seventh power - Wikipedia

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Result of multiplying seven instances of a number

In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n . 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]

Properties [ edit ]

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] 102 7 = 12 7 + 35 7 + 53 7 + 58 7 + 64 7 + 83 7 + 85 7 + 90 7 . {\displaystyle 102^{7}=12^{7}+35^{7}+53^{7}+58^{7}+64^{7}+83^{7}+85^{7}+90^{7}.} The two known examples of a seventh power expressible as the sum of seven seventh powers are 568 7 = 127 7 + 258 7 + 266 7 + 413 7 + 430 7 + 439 7 + 525 7 {\displaystyle 568^{7}=127^{7}+258^{7}+266^{7}+413^{7}+430^{7}+439^{7}+525^{7}} [7]

and

626 7 = 625 7 + 309 7 + 258 7 + 255 7 + 158 7 + 148 7 + 91 7 {\displaystyle 626^{7}=625^{7}+309^{7}+258^{7}+255^{7}+158^{7}+148^{7}+91^{7}} [7] 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.

What is power 5 called?

In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together: n5 = n × n × n × n × n. Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube.

en.wikipedia.org - Fifth power (algebra) - Wikipedia

The result of multiplying five instances of a number together

In arithmetic and algebra, the fifth power or sursolid[1] of a number n is the result of multiplying five instances of n together:

n5 = n × n × n × n × n .

Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube.

The sequence of fifth powers of integers is:

Properties [ edit ]

For any integer n, the last decimal digit of n5 is the same as the last (decimal) digit of n. I.E:

n ≡ n 5 ( mod 10 ) {displaystyle nequiv n^{5}{pmod {10}}}

By the Abel–Ruffini theorem, there is no general algebraic formula (formula expressed in terms of radical expressions) for the solution of polynomial equations containing a fifth power of the unknown as their highest power. This is the lowest power for which this is true. See quintic equation, sextic equation, and septic equation. 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. Specifically, 275 + 845 + 1105 + 1335 = 1445 (Lander & Parkin, 1966)[2]