Quadratic Reciprocity: The connection that changed number theory

https://www.quantamagazine.org/the-hidden-connection-that-changed-number-theory-20231101/

by Brajeshwar • 3 years ago

84 32 3 years ago

Edit: I corrected something in the original comment.

There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4.

We can generalize this. There are 2 kinds of primes with respect to mod 10. Those whose modulo is 2 or 5, and those whose mod x falls into some other odd number. Of the latter numbers, the more primes you consider the more it looks like a uniform distribution. This is related to the Riemann hypothesis [1].

But for the purposes of this article; we have that the mod is 4.

[1] https://www.youtube.com/watch?v=dwe4-OiRw7M

fatherzine• 3 years ago

Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.

lisper• 3 years ago

And while we are at it why is 1 not a prime? It conforms to the definition.

ducttapecrown2• 3 years ago

Because many formulas associated to a prime have a factor of p-1, so it's trivial in the case of p=2. There's also other weird stuff like no negatives and no quadratic formula in characteristic 2.

empath752• 3 years ago

Why do people say '2 is a solitary outlier'?

In this particular split of the prime numbers there are 3 categories, and 2 is the only member of one of the categories, while the rest are evenly divided. So it's a solitary outlier.

svat3• 3 years ago

You're replying to a comment that already answers your question. :-) I'd use slightly different words and say:

• When you work modulo 4 (as in the article in the context where this comes up), there are three kinds of primes: 2 is a solitary outlier, and all other primes are equally distributed between being 1 and 3 mod 4.

• When you work modulo 10, there are six kinds of primes: 2 and 5 are outliers, and all other primes are equally distributed between being 1, 3, 7 and 9 mod 10.

So 2 is an outlier when working mod 4 (or any even number). Mod 4 is crucially important in the context of quadratic reciprocity, so it matters here.

priprimer• 3 years ago

what I am amazed by is how of the primes congruent 1 mod 4 can be factored into complex numbers

but what puzzles me is that 11,13,17,19 are primes just like 101,103,107,109

but the real punchline is that 23 is the next prime after 19 AND 113 is the next prime after 109. WHY!?

this must be connected to 2*5

but I only have questions and confusion

joewferrara• 3 years ago

Gauss died in 1855 and his work still influences modern mathematics. Someone told me while in grad school that in number theory we’re still figuring out stuff that Gauss already knew.

keithalewis• 3 years ago

His motto was "Few, but ripe." I'm sure he had no conception of throwing compute power against the wall and see what sticks.

mmmmmmmike• 3 years ago

I think that’s more about what he chose to publish. He was known for discovering things and keeping them to himself.

As I understand it, he actually first came to conjecture quadratic reciprocity after doing incredible amounts of calculation by hand and noticing the pattern.

svat3• 3 years ago

Quadratic reciprocity is the first theorem I encountered in number theory for which I never developed an intuition for why it's true, and it's heartening to read in the article that professional mathematicians still feel the same way about it.

It can turn up unexpectedly. For instance, here's a problem/puzzle that a co-worker came up with: Find all positive integers b for which whenever (x^2 + xy + y^2) written in base b (for some integers x and y) ends with 0, it ends with two 0s. (Answer in rot13: gubfr gung ner n cebqhpg bs qvfgvapg cevzrf gung ner rnpu gjb zbqhyb guerr. This can be proved using quadratic reciprocity, at least the special case that (-3) is a square mod p iff p is a square mod 3.)

rpois• 3 years ago

https://en.wikipedia.org/wiki/Quadratic_reciprocity

Quadratic Reciprocity: The connection that changed number theory

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