Is there any such theorem that states this?

[Mr. P-1]

Originally Posted by soumya

One of my friends suggested a simpler proof which I want to share. Let's presume all factors of C are greater than 'i'. Therefore, all such factors can be written in the form i+k. Let's presume the smallest factor of 'i' is F, which is equal to i+x. So now we can say that C=(i+x) * (i+y) = i^2 +i(x+y)+xy, which is greater than i^2. So our first proposition must be false.

This is nonsense. Nothing forces you to conclude that the smallest factor of i is equal to i+x with x positive (which you didn't say, but which your proof requires.) If it did, then you could stop right there as you have the required contradiction.

The proof is fixable, by defining y differently: suppose C is composite and that i < (all proper factors of C) < C < i^2. Let A be a proper factor of C. Then so is C/A. Let A=i+x and C/A=i+y for +ve x and y. But then C = A*C/A = (i + x)(I + y) = i^2 + i(x+y) + xy > i^2. Contradiction.

A big "thank you" to all of you. But I would still want to know, if such a theorem already exists?

Of course it does. This is really trivial stuff you know.

[axn]

Originally Posted by Mr. P-1

This is nonsense. Nothing forces you to conclude that the smallest factor of i is equal to i+x with x positive (which you didn't say, but which your proof requires.) If it did, then you could stop right there as you have the required contradiction.

Looks to me like a simple typo where "smallest factor of 'C'" was intended. Wouldn't that "fix" the proof?

[Mr. P-1]

Originally Posted by axn

Looks to me like a simple typo where "smallest factor of 'C'" was intended. Wouldn't that "fix" the proof?

That, and saying that x must be positive, and defining y, and also saying that y must be positive.

So yeah, you can fix the proof by including everything he omitted. Also there's no reason for F to be the smallest factor of C. The proof works just as well if you say "let F be any factor of C". It's not a good idea to introduce redundant hypothesis into a proof. It may not make the proof technically "wrong", but it can obfuscate what is going on.

[NBtarheel_33]

Originally Posted by ATH

so .

At least one factor of C is <= and therefore < i which is what OP asked.

...and that's all you need. Multitasking while writing a proof and trying to show more than you need is a recipe for disaster...

[axn]

Originally Posted by Mr. P-1

That, and saying that x must be positive, and defining y, and also saying that y must be positive.

So yeah, you can fix the proof by including everything he omitted.

Hmmm... But your "nonsense" is unjustified. Could you tone it down a little? Another RDS, we don't need.

[Mr. P-1]

Originally Posted by axn

Another RDS, we don't need.

Would that I were as knowledgeable as he is.

[axn]

Originally Posted by Mr. P-1

Would that I were as knowledgeable as he is.

Ditto. It is the "other" stuff though...