For instance, imagine asking that same question about Euler's totient function 50 years before RSA was developed.
I am sure that there are countless other examples where a seemingly useless theorem aided in a practical problem decades or centuries later.
So is there really never a proof in pure mathematics with more obvious or immediate near-term applications?
Are there any historical examples of that? A major breakthrough in mathematics that once understood, it was immediately obvious that it was going to make X work better, and then it did?
Pure mathematics essentially concerns itself with the mathematics of mathematics. It's basically the process of trying to either prove certain "empirically discovered" mathematical facts, or understand deeply what those facts mean in a more general way. Understanding this type of mathematics in this deep way allows further mathematics to bud off of that specific