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Difference between revisions of "Perfect power"
 
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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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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 <math>64</math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th 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.
 
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.
  
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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.
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Perfect second powers are also known as [[perfect square]]s and perfect third powers are also known as [[perfect cube]]s.  This is because the [[area]] of a [[square (geometry) | square]] and the [[volume]] of a [[cube (geometry) | cube]] is equal to the second and third powers of a side length, respectively.

Latest revision as of 15:52, 18 August 2013

A positive integer $n$ is a perfect power if there exist integers $m, k$ such that $k \geq 2$ and $n = m^k$. In particular, $n$ is said to be a perfect $k$th power. For example, $64 = 8^2 = 4^3 = 2^6$, so $64$ is a perfect $2$nd, $3$rd and $6$th power.

We restrict $k \geq 2$ only because "being a perfect $1$st power" is a meaningless property: every integer is a $1$st power of itself.

Perfect second powers are also known as perfect squares and perfect third powers are also known as perfect cubes. This is because the area of a square and the volume of a cube is equal to the second and third powers of a side length, respectively.

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