Seventh power

{{Short description|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:

:{{math|size=120%|1=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:

: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, ... {{OEIS|id=A001015}}

In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid".{{r|womack}}

Properties

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{{r|dickson}} (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.{{r|kumchev}} If powers of negative integers are allowed, only 12 powers are required.{{r|choudhry}}

The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800.{{r|ekl}}

The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:{{r|stewart}}

: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 (M. Dodrill, 1999);{{r|meyrignac}}

and

:626^7 = 625^7+309^7+258^7+255^7+158^7+148^7+91^7 (Maurice Blondot, 11/14/2000);{{r|meyrignac}}

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.

See also

References

{{reflist|refs=

{{citation

| last = Choudhry | first = Ajai

| doi = 10.1006/jnth.1999.2465

| issue = 2

| journal = Journal of Number Theory

| mr = 1752254

| pages = 266–269

| title = On sums of seventh powers

| volume = 81

| year = 2000| doi-access = free

}}

{{citation

| last = Dickson | first = L. E. | author-link = Leonard Eugene Dickson

| doi = 10.2307/2301430

| issue = 9

| journal = American Mathematical Monthly

| mr = 1523212

| pages = 547–555

| title = A new method for universal Waring theorems with details for seventh powers

| volume = 41

| year = 1934| jstor = 2301430 }}

{{citation

| last = Ekl | first = Randy L.

| doi = 10.1090/S0025-5718-96-00768-5

| issue = 216

| journal = Mathematics of Computation

| mr = 1361807

| pages = 1755–1756

| title = Equal sums of four seventh powers

| volume = 65

| year = 1996| bibcode = 1996MaCom..65.1755E

| doi-access = free

}}

{{citation

| last = Kumchev | first = Angel V.

| doi = 10.1090/S0002-9939-05-07908-6

| issue = 10

| journal = Proceedings of the American Mathematical Society

| mr = 2159771

| pages = 2927–2937

| title = On the Waring-Goldbach problem for seventh powers

| volume = 133

| year = 2005| doi-access = free

}}

Quoted in {{citation

| last = Meyrignac

| first = Jean-Charles

| url = http://euler.free.fr/records.htm

| title = Computing Minimal Equal Sums Of Like Powers: Best Known Solutions

| date = 14 February 2001

| access-date = 17 July 2017

}}

{{citation

| last = Stewart | first = Ian | author-link = Ian Stewart (mathematician)

| isbn = 0-631-17114-2

| mr = 1253983

| page = 123

| publisher = Basil Blackwell, Oxford

| title = Game, set, and math: Enigmas and conundrums

| url = https://books.google.com/books?id=JRPdAwAAQBAJ&pg=PA123

| year = 1989}}

{{citation

| last = Womack | first = D.

| issue = 1

| journal = Mathematics in School

| pages = 23–26

| title = Beyond tetration operations: their past, present and future

| jstor = 24767659

| volume = 44

| year = 2015}}

}}

{{Classes of natural numbers}}

Category:Integers

Category:Number theory

Category:Elementary arithmetic

Category:Integer sequences

Category:Unary operations

Category:Figurate numbers

{{algebra-stub}}