The interestingness of a problem like this comes from its relatedness to other concepts in mathematics. The problems that are of interest are those whose solutions might contain concepts or truths that can be relevant to other fields of math. So a good math problem is not only something that is solvable and can lead to insightful truths about math, but also something that isn't too arbitrary or separated from the rest of math. Take the first question; it is about proving that the sequence of the central column isn't periodic. Whatever solution may come forth might also be relevant when proving or evaluating things related to prime numbers, which are equally a human construction based on a very simple rule, get equally complex equally fast, and are often looked at in terms of series of numbers. It may turn out to be a dead end, but it also may not, and the solution to this problem might just turn out to be transferrable to another set of problems.
That's what I expected, and what I was getting at with my attempt at recognizing that, in mathematics, the process of solving a problem can be even more valuable than the solution. Are they? I thought they were a basic element or fact of nature, that we just found an easy way to represent mathematically...? Like E=MC2... it is just a mathematical expression of a physical property. The Wolfram problem expressed in the article seems completely divorced from any practical, real world property. It's like he dropped six dice on a table, arranged them in some perceived order, and then made up some questions to ask about that order: Does it repeat? Is it infinite? What happens when you run the calculation a billion times? I'm mostly just marveling at how other people's brains work... not looking for an "answer", per se, just enjoying the window into other thought processes... "...The problems that are of interest are those whose solutions might contain concepts or truths that can be relevant to other fields of math...."
"...prime numbers, which are equally a human construction..."
Are numbers a fact of nature? You can count things, sure, but you'd never find just an idea of some number lying around. It's just as much of abstraction and tool as language or writing. The debate about maths (and, by extension, its study subjects like numbers) being invented or discovered is still ongoing. Philosophy aside, primes are just natural numbers that were observed to have some interesting property. Apart from being building blocks for numbers and them popping up everywhere in mathematics, there wasn't that much real-world use for them until the advent of modern cryptography (the 1970s or so). That's despite the fact people knew about and studied them since at least the times of Euclid. Fun and related to this thread story: classical electrodynamics, as formulated by Maxwell's equations, was compatible with special relativity good 40 years before special relativity was a thing. Even gave enough of a framework to derive Einstein's SR from them. It 'just' took someone pondering the right things to make it into its own thing.Are they? I thought they were a basic element or fact of nature, that we just found an easy way to represent mathematically...?
Like E=MC2... it is just a mathematical expression of a physical property.