Difference between revisions of "Perfect power"
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| − | A [[positive integer]] <math>n</math> is a '''perfect power''' if there | + | A [[positive integer]] <math>n</math> is a '''perfect power''' if there exist integers <math>m, k</math> such that <math>k \geq 2</math> and <math>n = m^k</math>. In particular, <math>n</math> is said to be a ''perfect <math>k</math>th power''. For example, <math>64 = 8^2 = 4^3 = 2^6</math>, so 64 is a perfect 2nd, 3rd and 6th power. |
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| − | For example, <math>64 = 8^2 = 4^3 = 2^6</math>, so 64 is a perfect 2nd, 3rd and 6th power. | ||
| + | We restrict <math>k \geq 2</math> only because "being a perfect <math>1</math>st power" is a meaningless property: every integer is a <math>1</math>st power of itself. | ||
Perfect second powers are usually known as [[perfect square]]s and perfect third powers are usually known as [[perfect cube]]s. This is because the [[area]] of a [[square (geometry) | square]] ([[volume]] of a [[cube (geometry) | cube]]) with integer [[edge]] is equal to the second (respectively third) power of that edge, and so is a perfect second (respectively third) power. | Perfect second powers are usually known as [[perfect square]]s and perfect third powers are usually known as [[perfect cube]]s. This is because the [[area]] of a [[square (geometry) | square]] ([[volume]] of a [[cube (geometry) | cube]]) with integer [[edge]] is equal to the second (respectively third) power of that edge, and so is a perfect second (respectively third) power. | ||
Revision as of 16:51, 18 August 2013
A positive integer 
is a perfect power if there exist integers 
such that 
and 
. In particular, is said to be a perfect 
th power. For example, 
, so 64 is a perfect 2nd, 3rd and 6th power.
We restrict only because "being a perfect 
st power" is a meaningless property: every integer is a st power of itself.
Perfect second powers are usually known as perfect squares and perfect third powers are usually known as perfect cubes. This is because the area of a square (volume of a cube) with integer edge is equal to the second (respectively third) power of that edge, and so is a perfect second (respectively third) power.
