Number theory prime number conjecture

Question

I came up with a theorem that a number $n$is prime if it is not divisible by any prime number $a \le \approx \sqrt{n}$

My proof is that past that set limit any prime number divisible would share a factor less than the same set limit, which should have already been discovered had it existed. Hope that makes sense.

Take for example 29. Approximately its square root is 5. It's not divisible by any primes up to 5. Therefore if it was divisible by say 7, then the corresponding factor i.e $7 \cdot k = 29$ would have to be less than 5. But it was never discovered from 2 upwards which is a logical contradiction.

Is there a way to write this down analytically?

Answer 1

Start by considering two factors of a number $n$ , let them be $a,b$ .

We will prove this using method of contradiction .

Assume $a,b\gt \sqrt n$ .

Then , $$a\times b \gt \sqrt n \times \sqrt n \implies ab\gt n$$

But this contradicts the fact that $ab = n$ . This is because of the incorrect assumption that both $a,b \gt \sqrt n$ holds at the same time.

Answer 2

It's a good observation, and it can be proven as follows:

Let $n=ab$, and let $a<b$. Suppose, for sake of contradiction that $a>\sqrt{n}$, then $$n=ab>\sqrt{n}\sqrt{n}=n$$ Therefore $a\leq \sqrt{n}$, and so the smallest factor is less than or equal to $\sqrt{n}$.

Now if no prime $p<\sqrt{n}$ divides $n$, you can conclude that $n$ cannot be written as $n=ab$, with $a,b$ non-units, which means $n$ is prime.