How are there unsolved math problems?
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Most proofs in math start with a conjecture. Some mathematician notices some pattern with a small set of numbers, then tests that pattern over more and more numbers. At this point they are just looking for a number or numbers that violate the pattern to prove the pattern doesn’t exist.
However, there are an infinite number of numbers so they can’t prove the pattern applies to every number.
Instead, proofs must cleverly combine what is known or proven already to prove the pattern.
For example, if you can prove that all bears are brown and prove that all grizzlies are bears, you can now make a proof saying that all grizzlies are brown.
But the proofs are much more complicated than this and there are many, many proofs you can use as steps in proving your conjecture. In our example, we could have a proof that proves that all fish swim. However, that proof doesn’t help prove a grizzly is brown… Or could it?
The sheer massive amount of proofs that can be combined and found to be applicable to each other in clever ways is nearly infinite too, so it’s not always easy to prove new things using old proofs.
Interestingly, people have started using computers to combine random proofs together to try and prove new things but this is obviously quite tricky to encode.
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