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}}
:
The two known examples of a seventh power expressible as the sum of seven seventh powers are
: (M. Dodrill, 1999);{{r|meyrignac}}
and
: (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=
| 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
}}
| 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 }}
| 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
}}
| 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
}}
| 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
}}
| 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}}
| 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:Elementary arithmetic
{{algebra-stub}}