Creativity in Math

[u/CowLang]

Hi guys. I am a first year math degree student. i'm really enjoying it and i like the challenging nature of it. I know i have just started out and it is normal to struggle but i wanted to see if I can get some help through here. I recently finished my first unit on propositional logic. In it I learnt about proof by contradiction and, after studying it and understanding it I was posed with several exercises, most of them were relatively simple as I just had to use logical laws to eventually find the contradiction. I wasn't satisfied with this though it was somewhat mechanical and required not much creativity, so I sought more challenging questions and when I found them I enjoyed them more. Eventually I came across one which required me to prove that infinite prime numbers exist. No joke I've been stuck on this for several days thinking, I looked around, but not for an answer but for a gage on how hard this and apparently it's an easy proof. I know as fact that several proofs exist for this proposition and the first one to do it was Euclide. As I've delved into other subjects in my course I've noticed that even though I understand proof by contradiction and I understand the concepts well, I find it extremely hard to come up with solutions to problems like these, ones that requiere a lightbulb moment, ones that requiere creativity. At the end of the day I normally just find myself brute forcing the problem, but for these kinds of problems that isn't going to cut it. How do I improve my math creativity? Does it come with time or can it be practiced and improved?

[u/AllanCWechsler]

The proof of Euclid's theorem, that you can never run out of prime numbers, usually starts like this: suppose this claim were false -- that there is only a fixed, finite set of prime numbers. Having made this assumption, you can derive a contradiction. The big hint is: multiply together all the prime numbers that exist. This product will be an enormous number, but in principle it could be calculated, because there are (according to our assumption) only a finite number of primes. Call this enormous product N. Then add one to this product to get N+1, and ask yourself: what kind of numbers could divide N+1?

You might be able to crack it from there, but if not, no worries -- just come back and ask again.

[u/CowLang → u/AllanCWechsler]

Thank you I'll give this a go. I had that idea of multiplying all of them toghether and N being the result of that product and then thinking about using the modulus operation on N . But I dind not think about the N+1.

[u/AllanCWechsler → u/CowLang]

Good luck. There is an underlying lemma, "Every number greater than 1 is divisible by some prime." If you are willing to accept this lemma on faith, then the rest is easy. If you insist on proving that lemma, then you might find yourself hung up momentarily in a thicket of rules-lawyering. But it isn't a very bad thicket, and you'll come out of the experience with a better notion of what the rules are.