
The Twin Primes Conjecture Gets Solved for Finite Fields - aburan28
https://www.wired.com/story/a-big-question-about-prime-numbers-gets-a-partial-answer/
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dvt
Actual paper:
[https://arxiv.org/pdf/1808.04001.pdf](https://arxiv.org/pdf/1808.04001.pdf)

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umvi
How can you prove a conjecture that concerns infinite numbers of primes using
a finite number system?

I'm not a mathematician, genuinely curious.

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dvt
> How can you prove a conjecture that concerns infinite numbers of primes
> using a finite number system?

Your intuition is correct, but doesn't go far enough! In fact, the notion of a
_prime_ in general doesn't make much sense (in Galois fields) :)

The article explains it fairly well. Basically, what was proved was the fact
that there exist an infinite number of twin prime polynomials (i.e.
polynomials that cannot be factored and differ by a fixed gap) in finite
number systems.

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AaronFriel
An interesting aside is that the encryption system most of us rely on every
day, AES, relies on a prime over a finite (Galois) field 2^8. The "subbytes"
step is a one to one mapping of a byte to another byte, and it's defined in
terms of a prime polynomial in the field GF(2^8). In practice though, the
table is pre-computed and hardcoded into AES implementations.

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TimTheTinker
> In practice though, the table is pre-computed and hardcoded into AES
> implementations.

It's often referred to as the "s_box" and the "reverse s_box" (for de-
encryption).

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Jeff_Brown
Is this of interest (so far) only to mathematicians, or does it have
applications in other fields (crypto?).

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kristianp
"Is there a cryptography application" was my thought too, however I'm guessing
pairs of primes aren't used there.

