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Comment by raverbashing | Hacker News Reader

Yes

Number Theory is a tough nut to crack and I feel that, while mathematicians have advanced a lot, there hasn't been any "master theorem" to handle it

Factorization, one way or another, still is done with brute-force (even if it's done in the EC domain or with NFS or related algos). Same for modular logarithm

Most of the theorems still fell like digging a pool with a spoon. Fermat's last theorem was finally proven, with a lot of effort and diversions. It really seems we're missing something "basic" (but complex) about it

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I'm not so sure that we're missing anything basic. The top rated answer by Scott Aaronson seems spot on to me. The natural numbers are just too powerful. As the resolution of Hilbert's tenth problem shows, we can at best hope to solve special cases of statements involving natural numbers, and the solvable cases are bound to require techniques of increasing complexity.

What would count as a "master theorem" for some field, anyway? I suppose something like Tarski's quantifier elimination of semialgebraic sets over R^n might be an example, though for actual computations the best algorithms have double exponential worst case complexity (i.e. even worse than integer factorization). Or for something even more basic, perhaps the fundamental theorem of linear algebra (along with Gaussian elimination). There's the saying that mathematics is the art of reducing any problem to linear algebra -- I suppose the hard problems in number theory are the ones that don't have a good linear algebra reformulation :-)

Yeah I read that comment and I agree with you

> What would count as a "master theorem" for some field, anyway?

I think you gave good examples, but it could be something like the discovery of Calculus, where suddenly a lot of problems became tractable. And in the end we even managed to solve (some) differential equations by solving polynomial equations.

So maybe you're right and it's linear algebra all the way down, but we're missing the weird trick to convert a factorization problem into a polynomial or something like that

Not a (master) theorem per se, but how about the classification of finite groups?

>Most of the theorems still fell like digging a pool with a spoon. Fermat's last theorem was finally proven, with a lot of effort and diversions.

The Fermat's last theorem equation was an incredibly difficult edge case that is illustrative of very fine boundary of a grand theorem. In the case of FLT, that was the modularity theorem.

I wouldn't call anything put towards proving FLT a diversion, the modularity theorem and other theorems that came as a result of proving FLT are the 'basic' (but complex) discoveries that these problems yield.

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