# Thread: Help to prove Primes

1. ## Help to prove Primes

Hello,

Can anyone please asssist with the following question:

Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the exponents of the primes are even integers.

I would surely appreciate it.

2. Originally Posted by lahuerita
Hello,

Can anyone please asssist with the following question:

Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the exponents of the primes are even integers.

I would surely appreciate it.
$p_1^{a_1}\cdot ... \cdot p_n^{a_n}$
You can write,
$\left(p_1^{a_1/2}\cdot ... p_n^{a_n/2} \right)^2$
Only when $a_i$ are divisible by two.
Thus, is $a_i$ are all even.

3. Thanks for the responds that clarifies a lot

4. ## Proving the reverse

How do you think specifically we could prove the only if part?

5. Originally Posted by lahuerita
How do you think specifically we could prove the only if part?
Simple, because "square" means there exists an integer $c$ such that,
$c^2=a$.
Now, no matter what canonical prime decomposition $c$ has $c^2$ MUST have even prime exponents because of the square (it doubles everything). And then you have that $a$ must also have even exponents because of uniqueness of prime power decomposition.