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A new generation of mathematicians pushes prime number barriers
- tromp 3y ago> The sieve of Eratosthenes comes alive in this animation, which shows multiples of each prime loping along the number line. A more accurate depiction would not have a bunch of prime curves starting from 0, but would have each one sprouting from its prime just when that prime is passed over by all existing prime curves.
- test77777 3y agoIf you just make up a number claim it’s prime and nobody disputes it, it’s prime apparently. I don’t think it’s really possible to have a very large prime number, because unless someone has tried every factor it’s really not prime yet, honestly that explains a lot about the elusiveness of the concept.
- akarve 3y ago> If you just make up a number claim it’s prime and nobody disputes it You can test if a number is prime in polynomial time, much faster than a sieve. There’s no need to test every divisor to know whether a number is prime or not. Algos like RSA generate large primes millions of times every day—-there’s nothing to take on faith.
- creata 3y agoDoesn't RSA typically settle for numbers that are probably prime?
- akarve 3y agoTechnically I think so, so there’s a tiny bit of faith for RSA but absolutely none for primality in general https://en.wikipedia.org/wiki/AKS_primality_test https://en.wikipedia.org/wiki/AKS_primality_test
- test77777 3y agoI’d argue that you’re just misunderstanding what makes a number prime. You can literally never be 100% sure a randomly generated number is or isn’t prime, it’s just the way numbers work.
- adgjlsfhk1 3y agoyou absolutely can. aks is a polynomial time deterministic primality check.
- test77777 3y agoOk then break all cryptography with that. It just proves my point you can SAY you can do it but in practice you can’t.
- ndsipa_pomu 3y agoCryptography is about finding large prime factors of a large number. That's a much harder problem than just determining whether a specific number is prime or not.
- test77777 3y agoI don’t understand you just say it was about finding factors, okay yeah that determines whether a number is prime. You’re not telling me anything I don’t know, I just disagree with your stance.
- sweezyjeezy 3y agoThe article starts by saying mathematicians want things to work for large numbers, but that doesn't really get to the crux of the issue with primes. Infinite sequences are ubiquitous in number theory, and in general it will be infeasible to have a test for inclusion in these sequences for large enough numbers. But often they have structure that we can use to characterise them very precisely - think about square numbers, it's easy to say what the trillionth square number is, what it's remainder when you divide by 13, etc. What makes primes hard, and also interesting, is that they seem to be extremely unstructured, we believe they behave like a kind of random number generator, even though they are clearly not random. In fact many of the theorems and conjectures mentioned in the article actually hinge on this. Random numbers are unpredictable on a small scale, but on a large scale they have very nice distributional properties, whereas more structured ones of similar growth rate will often have undesirable restrictions on them.
- credit_guy 3y agoYou should check the concept of "primality certificate". https://en.wikipedia.org/wiki/Primality_certificate https://en.wikipedia.org/wiki/Primality_certificate
- faceloss 3y ago[dead]
- dabeddabed 3y ago"...automorphic forms, which have their own version of the Riemann hypothesis." What's the Riemann hypothesis for automorphic forms?
- testless 3y agoTo automorphic forms, you can associate an L-function. There are similar conjectures about the zeros of those functions than for the Riemann Zeta function. https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis
- dabeddabed 3y agothanks!
- dabeddabed 3y agoNever mind I think it means the Ramanujan–Petersson conjecture