
Dreaming of prime numbers in short intervals - heinrichf
https://www.quantamagazine.org/kaisa-matomaki-dreams-of-primes-20170720/
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tarre
I still remember Kaisa telling in primary school, that Euclid's proof of
infinite number of prime numbers is often considered as the most beautiful
proof in mathematics.

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osivertsson
Interesting, can you give more context on when and where this was?

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tarre
We were both in a very small school in the countryside and when I was on 5th
and she on 6th grade, we had wonderful teacher, Harri Ketamo (Google him and
you'll end up wondering, what on earth he was doing in that school). Harri was
especially interested in teaching mathematics. As there was only about ten
pupils in the class, he had possibility to teach more advanced pupils further
covering among other things programming and prime numbers.

I don't remember if it was in school or after school, when we had with Kaisa a
debate over whether or not there was infinite number of prime numbers. The
next day or so she presented, that it has been proven and here it is. Back in
those days I didn't get it completely, but she was already there.

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mixedmath
I like many of quanta's articles on mathematicians and scientists, and I
appreciate their contributions to science journalism. I would like to add one
aspect that I think this article downplays, but which is understandable to a
large audience.

Kaisa and Maksym study multiplicative functions, i.e. functions which satisfy
f(ab) = f(a)f(b) if a and b are relatively prime. A big part in Kaisa and
Maksym's fundamental technique boils down to understanding completely the
behavior of f on small primes, and giving somewhat loose bounds on the rest.
Central to their success is their ability to make quantitative the intuitive
statement that "most numbers have lots of small primes as factors". This
required a few new ideas, but the nugget is quite simple, I think.

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brianberns
As a layman, I wonder about the connection between primes and random numbers.
Obviously, the primes are not actually random, but they seem to exhibit a lot
similarity to random numbers in that they are chaotic over short intervals,
but seem smoother and better behaved over long intervals.

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osivertsson
_Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in
Mathematics_ [1] is a nice read if that kind of question occupies your mind.

[1]
[https://en.m.wikipedia.org/wiki/Prime_Obsession](https://en.m.wikipedia.org/wiki/Prime_Obsession)

~~~
heinrichf
And Marcus du Sautoy's "The Music of the Primes: Searching to Solve the
Greatest Mystery in Mathematics" [1] is an even nicer one.

[1]
[https://en.wikipedia.org/wiki/The_Music_of_the_Primes](https://en.wikipedia.org/wiki/The_Music_of_the_Primes)

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mudil
Can anyone explain to me why primes are so important in cryptography?

I understand that they are used for factorization, where it is easy to
multiply two primes and get a number, but it is difficult from a big number to
find what two primes were used to get it. So, the question is why not to have
a big database of primes, and when we have that big factorized number we try
to divide that number by each prime from the database and see if the result is
another prime from the database? Wouldn't that work?

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mixedmath
It is standard to take two primes that each have at least 100 digits or so.
There are approximately 10^100 / 230 primes of of this size (this is the Prime
Number Theorem). There is simply not enough space or time to store or
interpret 10^100 numbers.

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dalore
yes but you don't store the number, you store the index to the prime and a
function to generate it.

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mixedmath
The parent asked why we don't just search through all candidate primes.
Phrased differently, my response is that there are too many. This remains true
if you replace "primes" by "indices" \--- there are still too many.

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seycombi
Why prime numbers are usefull to Web Designers [https://www.sitepoint.com/the-
cicada-principle-and-why-it-ma...](https://www.sitepoint.com/the-cicada-
principle-and-why-it-matters-to-web-designers/)

~~~
johansch
Gah.

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sevenfive
Their example uses log_10 instead of ln

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impendia
(Professional mathematician and analytic number theorist here.)

This is not correct. When mathematicians (at least analytic number theorists)
use the notation "log", it always means to base e, i.e., ln.

Natural logarithms come up all the time in math, and in analytic number theory
in particular; base 10 is an artifact of how many fingers we evolved with.

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sevenfive
No, I mean they had a chart in the article that says Log(10) = 1. Looks like
it's been removed.

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impendia
Aha. Looks like the editors made (and caught) a mistake then.

