Coming up with short “magical” proofs

I was reading the solution to this problem:

Prove that $f(n) = 2n$ is the only non-constant solution to $2f (m^2 + n^2 ) = (f (m))^2 + (f (n))^2 .$

The solution used these identities, pulled out of the blue:

$(5k + 1)^2 + 2^2 = (4k + 2)^2 + (3k − 1)^2 ,$

$(5k + 2)^2 + 1^2 = (4k + 1)^2 + (3k + 2)^2 ,$

$(5k + 3)^2 + 1^2 = (4k + 3)^2 + (3k + 1)^2 ,$

$(5k + 4)^2 + 2^2 = (4k + 2)^2 + (3k + 4)^2 ,$

$(5k + 5)^2 + 0^2 = (4k + 4)^2 + (3k + 3)^2 .$

and proceeded to use strong induction to prove the hypothesis.

As I have spent more and more time studying mathematics, I see more of these "magic" solutions in which some obscure identity or property is pulled out of nowhere and used to facilitate a proof...often, multiple such jumps are used in a proof...and these proofs are meant to be done in an hour or so. I feel discouraged by this, because I don't understand how to do these "magical" things. I feel like I could spend 10 years trying to solve this problem, and still fail, because I don't have magical powers.

How can I improve myself? How do I learn to come up with magical proofs like these? I feel like practicing problem solving is useless because I can solve all the easy ones...but the moment a "hard" problem comes, I can't solve it at all. So I read the solution, and understand it, but the next problem is completely different and uses a different magical identity and so I haven't actually learned anything.

I'm sorry for ranting here, but I feel hopeless. I'm 19 and some people my age are solving these problems easily. Do I lack mathematical ability genetically or is is something I can gain through practice? I am wondering if I am just wasting my best years studying mathematics when I don't have the genes for it.

EDIT: For those asking why I couldn't just use a substitution...the actual problem was . to find all solutions, not just $f(n)=2n$

• Hey now--I'm 19 and I'm nowhere near close to solving that problem. ;) The important thing to keep in mind is that, no matter your skill level, there will always be someone better and someone worse than you in at least one topic. Don't focus on being the best--focus on doing your best and being the best comes naturally. – apnorton Jul 19 '14 at 22:46
• I don't see why anyone would solve this by citing these identities. You can just evaluate both sides with $f(n) = 2n$ and see that they come out the same. – Jair Taylor Jul 19 '14 at 22:57
• Thank you so much for not using the word 'mathemagic' in your post – enthdegree Jul 19 '14 at 22:59
• @Ethan Thank you for changing the title. I've added the tags soft-question and proof-writing, as I think they describe more accurately the idea of the question. – Mark Fantini Jul 19 '14 at 23:05
• I think you should not attempt to train yourself to produce magical proofs. As you have already seen, other people learn very little from these. You should train yourself to produce understandable proofs that teach people things. – Qiaochu Yuan Jul 19 '14 at 23:48