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Comment by Yajirobe | Hacker News Reader

So if RH is proven, what actually changes? As far as I know, there are tons of theorems that already presuppose RH to be true There wouldn't suddenly be an insight into how to find larger primes, for example.

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The method used to solve RH would almost certainly contribute substantially to our understanding of mathematics and give new directions to tackle other questions in mathematics. For example, the resolution of the Poincare Conjecture by Perelman involved making progress with the technique of "Ricci flow" which is still an active area of research both in itself and for tackling other problems. The Poincare Conjecture per se is often not involved in these Ricci flow questions even though the proof of the conjecture is used.

Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axioms, and he wanted to ensure everything was built on a logical foundation and was rigorous.

In practice it may not change much, but more philosophically speaking, it's dangerous to take something as granted without a proof. If we then have many ideas building off this unproven idea, and it's later shown to be false, a lot of those theorems would be based off a false premise. This doesn't necessarily mean they the proposition is incorrect, but it would mean that they are unproven.

It's more to satisfy the rigour in maths, and also pose as a challenge to mathematicians. In the journey to proving RH a lot of new ideas could be found that could prove useful, too.

Proving some things may become easier (as an example of that there are already more than a few papers with proofs that rest on RH ... and then additional effort to go and find RH-less proofs.

A silly example (from memory) using another rather non-obvious proof.

1. Assume 2^(1/4) = X/Y for integers X/Y (Assume the quartic root of 2 is rational)

2. Raise both sides to the 4th power: 2 = X^4 / Y^4

3. Multiply both sides by Y^4: Y^4 + Y^4 = X^4

4. QED, quartic root of 2 is irrational, by contradiction with Fermat's last theorem.

It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have confirmation that they are reality then. What would change is that we would likely have novel and powerful methods that were used in the proof. This is what most of the buzz is about. For example, the proof of Fermats Last Theorem introduced novel connections between sub fields of mathematics.

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