system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Why are there still unsolved math problems? Math is concrete, is it just that mathematicians don't know what steps to take?
gpt
Unsolved problems are not just arithmetic problems no one can find the answers to. They are usually conjectures about some mathematical structure or group of structures, and they are usually impossible to solve by brute force because the structures involved are usually infinite. For example:

* The Collatz conjecture. Pick any positive integer. If it's even, divide it by 2; otherwise, triple it and add 1. The Collatz conjecture is that if you repeat that process long enough, you will eventually reach 1. For example, if you start with 3 you'll go through 10, 5, 16, 8, 4, and 2 before reaching 1. There are two infinities here. First, the number can take an arbitrarily long time to reach 1 - as long as it gets there eventually. Try starting with 27 and see how long it takes to get to 1. Second, the conjecture is about *every* positive integer - of which there are infinitely many.
* Another long-standing open problem involves perfect numbers. If you take all the numbers that divide 6 (except 6 itself) and add them together, you get 1+2+3 = 6. Similarly, if you add the divisors of 28 you get 1+2+4+7+14 = 28. Numbers like these, which are equal to the sum of their divisors, are called "perfect". It is conjectured that there are no odd perfect numbers. But there are infinitely many odd positive integers, so again this problem is about an infinitely large collection of things.
* A lot of people are surprised to find out that it is currently unknown whether pi+e is irrational. But "irrational" means it isn't equal to the ratio of two integers, and there are infinitely many possible ratios. There is no easy way of checking whether an arbitrary real number is rational.

A proof of any of these would have to involve more than just basic arithmetic operations and checking cases. You would have to prove something fundamental about the way that the divisors of integers work, for example, in order to show that no integer can be both odd and perfect. Many fundamental results have been found and proven already, but there's no single method that can resolve every question you'd ever want to ask about the integers (or other complicated infinite structures).