Proving that there exists something. When asked to "Prove that there exists such x that y"  , is giving such "x" enough as a solution or do you need to find like a general formula or something? For example, if asked to "prove that there exists n ≤ d such that d|2^(n)−1". One solution would be when n is 6 and d is 9. Help would be much appreciated. 
 A: If you’re asked to prove, say, that there are consecutive integers $a,b$, and $c$ such that $a^2+b^2=c^2$, then you’re being asked for just one thing, and producing one example ($a=3,b=4$, and $c=5$) is good enough. But in your earlier question you’re not asked to prove simply that there is an $n\le d$ such that $d\mid 2^n-1$: you’re asked to prove that for each odd positive integer $d$ there is an $n\le d$ such that $d\mid 2^n-1$. It’s not enough to provide one example; you have to provide at least one example for each odd positive integer $d$.
A: The question you've posted is imprecise: implicitly, it is asking you to prove that for all positive integers $d$, there exists an $n\leq d$ with $d\ \vert\ 2^n-1.$ In this case you don't get to pick the $d$. You need a proof that such an $n$ exists no matter what $d$ happens to be: a formula or algorithm that takes in $d$ and calculates $n$, for instance, would be enough.
A: This is a very good question, because you are starting to ask:  "what is a proof anyway?". If you ever are at a party full of mathematicians, just ask that -- everyone will start talking. However, for your situation we can be less general:
What you have to show depends on what is asked.  If you are to show that there exists a "Q" which something or other, then you only need to demonstrate that a Q exists.  You might do that by 
a) producing a specific Q
b) giving a formula or algorithm which would produce such a Q
c) showing that if there isn't such a Q something inconsistent is true, like 2 = 1.
d) showing indirectly that there must be a Q, even if you haven't a clue how to construct it.  
A fine example of this last idea is the proof that a polynomial of degree n has n roots.
This is quite distinct from showing a "for all".  You might be asked to show that for any infinite series $\sum_0^{\infty}a_n$  which converges, then $a_n \rightarrow 0$.  It wouldn't help at all to show that this is true for one specific series, or even lots of specific series. So you have to come up with a general argument that covers every convergent series.
General arguments such as that depend very heavily on the reading the hypotheses very carefully.  A small change in the hypotheses could make the result untrue.  Then you follow Dan Fisher's advice and produce a counterexample.  This falls back into the case of showing there is a Q (for which the hypotheses do not produce the desired conclusion).
Don't overlook other possibilities.  Most serious math problems are not solved instantly.  People chip away at them and get partial results.  Even a partial result can be quite important and compelling.  For example, the twin prime conjecture has recently been bounded by an interval of 78 million; then that was reduced to about 8,000; and last I heard someone got it down to around 300.  They are still a ways from 2, but the partial results are important.
For purposes of being a good little student, a partial result is very useful:
a.  It might get you partial credit
b.  Better yet, it might give you insight into the problem and allow you to go on to a complete proof. 
