Square-free integer

From Simple English Wikipedia, the free encyclopedia

A square-free integer is a number which is not divisible by any square numbers other than 1. In other words, each prime number that appears in its prime factorization appears exactly once.

For example, is square-free. However, is not square-free: it is divisible by , and the power of in the prime factorization is to a power larger than one.

Möbius function[change | change source]

The Möbius function is a function which takes in natural numbers and is usually written as . The value of depends on whether or not is square-free. Specifically,

For example, is square-free with an 2 prime factors, so . Since is not square-free, then .

Since is prime, it is its own prime decomposition. That is, the prime factorization of has 1 prime factor, so .

References[change | change source]