TITLE: Is a counterexample considered a rigorous proof that a property is not true?
QUESTION [14 upvotes]: This is my follow-up question to my own query earlier:

How can I algebraically prove that $2^n - 1$ is not always prime?

Almost half of the answers said that I provided my own proof by giving the counterexample. Although, I was not satisfied through those answers since they didn't provide a complete view of the situation and how there exist algebraic factors.
Is disproving an algebraic statement necessarily the same as introducing a case where it is not true? I agree with Nameless's statement found here, but I am not sure as to how that contributes to a rigorous algebraic  proof.
Remark: I agree that I didn't ask for an algebraic proof earlier, but I did expect a better proof.

REPLY [8 votes]: I'd like to add a small remark to the other (excellent) answers, because it touches on what I think is the source of your discomfort with the proof by counterexample. As already noted, the negation of a universally quantified formula is logically equivalent to the existentially quantified negated formula.
And explicitly providing an example is one way to prove an existential statement. (So the following is not specific to counterexamples.) 
The key point here is that for this it is not enough to just give an object ("$n=4$"), but one needs to prove that this object indeed has the desired properties ("Then $2 ^4-1 = 15$ is not prime, since it is divisible by $3$"). The elementary nature of this concrete example (and some of the language used when talking about a proof by (counter)example) can hide this, but it is what makes it a proof (and can in fact be quite involved). 
(And if I am allowed to give some advice: If asked for an example - especially on a test, - never just state one, but always explain why it is one, even if it just takes a half-sentence.)