Trying to do a non-rigorous direct proof

[ull]

Statements:
x is an integer
x is a prime number if x doesn't consist of any prime factors ≤√x

Proof:
Since (√x + 1) * (√x + 1) > √x * √x
x must be a prime

Questions:
Whould you consider this a non-rigorous direct proof?
If not, what does it lack?
Is this a good approach trying to prove it?

The proof was meant to be like this:
Since √(x + 1) * √(x + 1) > √x * √x
x must be a prime

 

[Number Nine]

Recall that every integer >1 has a (unique) prime factorization. Now, suppose that ##x## is not prime, and further has no prime factor ##p_1 \leq \sqrt{x}## (so that ##p_1 > \sqrt{x}##); then ##x## has factorization ##x = p_1?##. What possible values can ##?## take? (Hint: what happens if ##? > \sqrt{x}##)

 

[ull]

Please don't post any more clues just jet I am trying to figure it out it :):)
I don't know if this translates very well, but: "I´m going to sleep on it " ;)

 

[verty]

I know you said no more hints, but this hint is just too useful not to suggest.

Hint: write the smallest divisor (not counting 1 of course) as ##\sqrt{x} + ε##.

Hmm, I wonder if this counts as a direct proof, probably not.

 

[jbriggs444]

It is always a good idea to start with a proper statement of what it is you are trying to prove.

"For all positive integer x, if x has no prime factors less than or equal to its square root then x is prime"

One problem with this formulation is that it is false. The positive integer 1 is a counter-example.