system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5- Why are there unsolved mathematical equations? Who wrote them and how do they know if they are solvable or not?
gpt
So...you should note that mathematical problems and mathematical equations aren't inherently the same thing.

An equation is basically just two sets of algebraic quantities that have an equal/greater than/less than sign between them. Some of these equations are inherently not solvable because they have self-contradictory answers. For example, let's say you have the equation;

1/(x + 3) + 1/(x - 3) = 6/(x^2 - 9)

You can manipulate it (multiply both sides by "(x + 3)\*(x - 3)") to get the following;

x + 3 + x - 3 = 6

or;

2\*x = 6

or;

x = 3

But that doesn't actually work, because if you plug x = 3 back into the original equation one of the two terms on the left side of the equals sign gives you a 1/0 situation, which means that x = 3 is essentially not "allowed" to be an answer. Thus, the equation has no solution, making it unsolved.

For unsolved *problems*; those are basically just mathematical statements that we can't quite figure out if they're true or not. For example, there's the Collatz Conjecture; pick a number, and then perform the following operations;

1) If it's even, divide it by 2.

2) If it's odd, multiply it by 3 and add 1.

Then feed the resulting number back in, and again, and again. Doing this, you'll (probably) end up with a repeating sequence of 4, 2, 1, 4, 2, 1, etc. No matter what your starting number is, you always seem to end up in this pattern because *eventually* you wind up with some number 2^N and the whole thing collapses.

The problem is that we can't figure out how to prove that it's explicitly true for all numbers; we can only observe that it hasn't been disproven yet, which is not good enough for mathematics. Thus, the problem is unsolved.