Seventh power

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]

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.

• Eighth power
• Sixth power
• Fifth power (algebra)
• Fourth power
• Cube (algebra)
• Square (algebra)

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