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And because this is HN, based on past experience, I need to preface with the fact that I'm a professor of math. So no need to start by questioning my knowledge on the topic, just get straight to the point.
Earlier quoted context omitted.
They’ve got the fundamental theorem of arithmetic. What more could they want?
I think that misses the point which is that the simplicity is overlooked in the common descriptions of primes as "random" or a great "mystery".
Analogously, if someone proves that P = NP, then that will be great, but the significance of lambda calculus and Turing completeness will remain. If the proof is constructive and practical, we’ll just have to reprioritize and update the list of algorithms we teach to undergrads, issue performance-enhancement updates to some software libraries, and patch any security vulnerabilities. Otherwise, we’ll only need to change a chapter or two in the Theory of Computation courses that universities are increasingly deprioritizing.
Earlier quoted context omitted.
does the move to elliptic crypto suggest that the people in the know expect prime factorisation to be broken soon?
Since no one mentioned, a major reason to prefer elliptic curves is that you can get equivalent security for much smaller key sizes.
Earlier quoted context omitted.
does the move to elliptic crypto suggest that the people in the know expect prime factorisation to be broken soon?
It's to do with "if we ever have big/good quantum computers, prime factorization is doable" and with "let's use the new shiny things". On a related note, someone might discover some elliptic curve math tomorrow and your CPU can break all that stuff just as well...
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That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question. - Gauss
Not only does these algebras give a more intutive understanding of "imaginary" numbers as rotation in a plane (and not simply an alternative R2).
They also extend nicely into all sorts of applications in Physics, Machine learning, etc where Lie groups are needed.
And there is nothing preventing us from defining e1*e2 as i, and use the regular notation for Complex Analysis where the Group Theory aspects are not needed.
This got me thinking. Imagine this discovery led to a larger breakthrough on prime numbers that allowed easy factorization of large integers and effectively rendered public key cryptography such as RSA ineffective overnight, by allowing anyone with a consumer-grade CPU to crack any production-size key. Does the industry have DR plans for this scenario? Can the big players quickly switch to a different, unbroken encry…
The ability to prepare for catastrophic scenarios is overestimated. The ability to survive them is underestimated.
Earlier quoted context omitted.
The world has moved away from RSA etc to elliptic curves. Not everybody did, through. RSA is no longer the standard, and has not been for many years.
Still, plenty of old stuff was scraped/sniffed under the "store now, decrypt later" methodology.
To see what I am talking about as in trivial and non-trivial zeros see this wikipedia animation https://en.wikipedia.org/wiki/File:Riemann3d_Re_0.1_to_0.9_I...
Basically, it implies that there is another relationship between real and imaginary numbers we have not yet stumbled upon....
And,this has implications upon finding the gravity theory as Riemann math is involved in quantum mechanics....
Strange science that primes is or might be involved in gravity theory....
This got me thinking. Imagine this discovery led to a larger breakthrough on prime numbers that allowed easy factorization of large integers and effectively rendered public key cryptography such as RSA ineffective overnight, by allowing anyone with a consumer-grade CPU to crack any production-size key. Does the industry have DR plans for this scenario? Can the big players quickly switch to a different, unbroken encry…
From what I remember in my math class where we made those cryptography calculations by hand, the teacher many years ago said that the day we could guess prime numbers it will be a disaster because many cryptographic calculations are based on the premise that we can’t guess prime numbers. I don’t know if that changed ?
This is from May and there was a better article in Quanta already discussed here. https://www.quantamagazine.org/sensational-proof-delivers-ne...