# 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:

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, ... (sequence A001015 in the OEIS)

In the archaic notation of Robert Recorde, the seventh power of a number was called the "second sursolid".[1]

## 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[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]

$\displaystyle{ 102^7=12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7. }$

The two known examples of a seventh power expressible as the sum of seven seventh powers are

$\displaystyle{ 568^7 = 127^7+ 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 }$ (M. Dodrill, 1999);[7]

and

$\displaystyle{ 626^7 = 625^7+309^7+258^7+255^7+158^7+148^7+91^7 }$ (Maurice Blondot, 11/14/2000);[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.

## References

1. Womack, D. (2015), "Beyond tetration operations: their past, present and future", Mathematics in School 44 (1): 23–26
2. "A new method for universal Waring theorems with details for seventh powers", American Mathematical Monthly 41 (9): 547–555, 1934, doi:10.2307/2301430
3. Kumchev, Angel V. (2005), "On the Waring-Goldbach problem for seventh powers", Proceedings of the American Mathematical Society 133 (10): 2927–2937, doi:10.1090/S0002-9939-05-07908-6
4. Choudhry, Ajai (2000), "On sums of seventh powers", Journal of Number Theory 81 (2): 266–269, doi:10.1006/jnth.1999.2465
5. Ekl, Randy L. (1996), "Equal sums of four seventh powers", Mathematics of Computation 65 (216): 1755–1756, doi:10.1090/S0025-5718-96-00768-5, Bibcode1996MaCom..65.1755E
6. Game, set, and math: Enigmas and conundrums, Basil Blackwell, Oxford, 1989, p. 123, ISBN 0-631-17114-2
7. Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions".