# Square-free number

From Groupprops

*This article defines a property that can be evaluated for natural numbers*

## Contents

## Definition

A natural number is said to be **square-free** if there is no prime number for which divides .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

cyclicity-forcing number | every group of that order is cyclic | see classification of cyclicity-forcing numbers | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

solvability-forcing number | every group of that order is solvable | square-free implies solvability-forcing | any square of a prime is a counterexample | |FULL LIST, MORE INFO |