Seventh power
In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So:
- n^{7} = 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]} All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers.^{[3]} If negative powers 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
- (M. Dodrill, 1999);^{[7]}
and
- (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.
See also
References
- ^ Womack, D. (2015), "Beyond tetration operations: their past, present and future", Mathematics in School, 44 (1): 23–26
- ^ Dickson, L. E. (1934), "A new method for universal Waring theorems with details for seventh powers", American Mathematical Monthly, 41 (9): 547–555, doi:10.2307/2301430, MR 1523212
- ^ 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, MR 2159771
- ^ Choudhry, Ajai (2000), "On sums of seventh powers", Journal of Number Theory, 81 (2): 266–269, doi:10.1006/jnth.1999.2465, MR 1752254
- ^ 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, MR 1361807
- ^ Stewart, Ian (1989), Game, set, and math: Enigmas and conundrums, Basil Blackwell, Oxford, p. 123, ISBN 0-631-17114-2, MR 1253983
- ^ ^{a} ^{b} Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 17 July 2017.
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