Cfrgtkky

I distinctly remember from my long-ago undergraduate math that there were some interesting cases where a solution (proof) was sought for some specific thing but it wasn't easy to find - and in a few of these cases a generalization was made to some wider problem and that turned out to be provable (maybe not easily, but understandably) and of course that in turn solved the specific case.

But I don't remember any such situation and would like to know of one or more.

(I don't think the whole issue of the power of complex numbers - e.g., in the Fundamental Theorem of Algebra or in complex analysis vs real analysis is what I'm thinking of. I'm sure there were more closely focused cases of this phenomenon that I once knew.)

Here's a very common type of example. Suppose you are interested in a certain convergent series $sum_{n=0}^infty a_n$. You look instead at the more general series $sum_{n=0}^infty a_n z^n$, to which you can apply the machinery of calculus, differential equations etc.,
and then specialize your result to $z=1$.

1. There are many examples where this happens with properties concerning integers, where a property for a given integer may not be easy to prove, but the proof for the $n=0$ or $1$ case is easy, and induction is easy as well.

Alain Connes gave a great example of this on the radio you could even explain to a non-mathematician (actually that was the point): if I give you a chocolate bar with $8times 5$ squares, and you're allowed to cut it by breaking any piece into two pieces along a straight line (not intersecting the squares), what is the way of doing it that requires the least amount of cutting ? For $8times 5$ it's not clear, but if you generalize to $ntimes m$ and use induction, then it becomes quite clear.

Perhaps a more mathematical example of this phenomenon would be quite appreciated: many divisbility questions fit into this, e.g. prove that $2^{2n}+2$ is divisible by $3$

2. A different family of examples comes from applications of group theory or ring theory to number theory; for instance Euler's theorem : $a^{varphi(n)} = 1 mod n$ if $aland n = 1$. I have never seen Euler's original proof of this, but seeing how complicated non-group-theoretic proofs of Fermat's little theorem are, I can only imagine that proving this only by number theory must be hard; while Lagrange's theorem is really easy and deals with this immediately if you know Bezout's theorem.

Another example in this family could be the fact that $nmid varphi(a^n-1)$; or $n! mid displaystyleprod_{i=0}^{n-1}(q^n-q^i)$ whenever $q$ is a prime power (though for these, unlike with the first family of examples, it may not be clear how to go from the specific instance to the generalization)