Square Free Integers

Square Free

An integer

f \in ℤ^{\neq 0}

is said to be square free if there is no

k \in ℕ_{1}

such that

k^{2} | f

Square Free Division Characterization

f \in ℤ^{\neq 0}

is square free iff for any

k \in ℤ

k^{2} | f ⟹ k \in {1, - 1}

Divisor of Square Free Product is Divisor of Other

Suppose that

f \in ℤ^{\neq 0}

f

is square free then

a^{2} | f b^{2} ⟹ a | b

for any

a, b \in ℤ

If a Square Free Number Divides a Square then It divides the Square Root

Suppose

f \in ℤ^{\neq 0}

, if

f

is square free then

f | a^{2} ⟹ f | a

Suppose that

f | a^{2}

, then we have

f^{2} | f a^{2}

therefore we have that

f | a

If a Square Free Number Divides the Nth Power then it Divides the Nth Square Root

Suppose that

f \in ℤ^{\neq 0}

is square free then

f | a^{n} ⟹ f | a

for any

n \in ℕ_{1}

We assume that

f | a^{n}

, now if

n = 1

we are done, otherwise

n \geq 2

and therefore we know that

2 n \geq 2 + n

so that

2 n - 2 \geq n \geq 2

, thus we have that

f | a^{2 (n - 1)}

therefore by assumption we know that

f | a^{n - 1}

, if

n - 1 = 1

we are done, otherwise

n - 1 \geq 2

and we repeat this process finitely many times until

n - j = 1

at which point we've deduced that

f | n

as needed.

Power of a Square free Number Divides the Power of a Number Implies the Square Free Number Divides the Other

Suppose

f \in ℤ^{\neq 0}

is square free, then

f^{f} | a^{a} ⟹ f | a

for any

a \in ℕ_{1}