A new generation of mathematicians pushes prime number barriers
quantamagazine.org
A new generation of mathematicians pushes prime number barriers
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Re: A new generation of mathematicians pushes prime number barriers
#2"...automorphic forms, which have their own version of the Riemann hypothesis." What's the Riemann hypothesis for automorphic forms?
Re: A new generation of mathematicians pushes prime number barriers
#3"...automorphic forms, which have their own version of the Riemann hypothesis." What's the Riemann hypothesis for automorphic forms?
Never mind I think it means the Ramanujan–Petersson conjecture
Re: A new generation of mathematicians pushes prime number barriers
#4"...automorphic forms, which have their own version of the Riemann hypothesis." What's the Riemann hypothesis for automorphic forms?
To automorphic forms, you can associate an L-function. There are similar conjectures about the zeros of those functions than for the Riemann Zeta function.
Re: A new generation of mathematicians pushes prime number barriers
#5> 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.
Re: A new generation of mathematicians pushes prime number barriers
#6> 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.
That would start to resemble some sort of cool fractal.
Re: A new generation of mathematicians pushes prime number barriers
#7If 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.
Re: A new generation of mathematicians pushes prime number barriers
#8If 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.
> 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.
Re: A new generation of mathematicians pushes prime number barriers
#9If 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.
The 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.
Re: A new generation of mathematicians pushes prime number barriers
#10> 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.
That would start to resemble some sort of cool fractal.