Seventh power

Inarithmeticandalgebratheseventhpowerof anumbernis the result of multiplying seven instances ofntogether. So:

n⁷=n×n×n×n×n×n×n.

Seventh powers are also formed by multiplying a number by itssixth power,thesquareof a number by itsfifth power,or thecubeof a number by itsfourth power.

Thesequenceof seventh powers ofintegersis:

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,... (sequenceA001015in theOEIS)

In thearchaic notationofRobert Recorde,the seventh power of a number was called the "second sursolid".

Properties

Leonard Eugene Dicksonstudied generalizations ofWaring's problemfor seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers(1⁷is 1, and 2⁷is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers.If powers of negative integers are allowed, only 12 powers are required.

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

The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:

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

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);

and

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

any example with fewer terms in the sum would be acounterexampletoEuler's sum of powers conjecture,which is currently only known to be false for the powers 4 and 5.

See also

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