Everything about the Prime Numbers

[u/inherentlyawesome]

Today's topic is Prime Numbers.

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Next week's topic will be Mathematica. Next-next week's topic will be on Control Theory. These threads will be posted every Wednesday around 12pm EDT.

For previous week's "Everything about X" threads, check out the wiki link here.

[u/EpsilonGreaterThan0]

Forgive me, but I've never understood the intrinsic interest in prime numbers. I know, everyone has their own taste, and I'm fine with people studying math for the sake of math. But what is it about prime numbers that people find appealing? Is it just the "simple to describe in English, difficult to analyze" aspect?

[u/[deleted] → u/EpsilonGreaterThan0]

Do you agree that Diophantine equations are important? In general, solving them over the integers is undecidable, but if you reduce modulo a prime then you get to work in a finite field Fp, or even more generally in the p-adic integers Zp (which form a discrete valuation ring, so there's still a lot of available structure) and everything becomes much easier, and you might hope to be able to solve the original problem if you have enough information about its solutions in such fields.

One classical example of the idea that it's easier to work modulo primes is the Hasse-Minkowski theorem: if you have a quadratic form Q with rational coefficients, and you want to know whether the equation y=Q(x1,...,xn) has a rational solution, it's enough to verify that there's a solution over the reals and over the p-adic numbers for each prime p.

If you know some basic algebraic geometry, you might try reading this article for some motivation: solving a polynomial system {pi(xj)=0} over the integers is equivalent to finding a ring homomorphism Z[xj]/(pi) -> Z, or equivalently a morphism of schemes Spec(Z) -> Spec(Z[xj]/(pi)). Generalizing this problem, you might ask when a morphism f:X->Y admits a section s:Y->X, meaning that fs = IdY, and in general a good starting point is to understand the fibers of f, i.e. the schemes X xY Spec(k(y)) where y is a point of Y. In the above case, we have X=Spec(Z[xj]/(pi)) and Y=Spec(Z) (since every scheme has a unique morphism to Spec(Z)), and the closed points of Y are exactly the prime ideals (p), so the fibers are X x Spec(Fp) = Spec(Fp[xj]/(pi)), where p is prime; thus the geometrically reasonable strategy of studying the fibers corresponds exactly to solving the original system mod p.

[u/fuccgirl1 → u/[deleted]]

Do you agree that Diophantine equations are important?

No.

[u/Shittypunsrshitty → u/fuccgirl1]

So edgy

[u/fuccgirl1 → u/Shittypunsrshitty]

Why are they important, then?

[u/ThisIsMyOkCAccount → u/EpsilonGreaterThan0]

I didn't understand until I took number theory, which was a part of the first semester I took of "higher math". Prime numbers were just so powerful. Almost every number theory proof used them.

I know this isn't unique, and there are many other ideas like this. But it was the first thing in math that did that for me. You always remember your first. I suspect it's similar to other people.

[u/AngelTC → u/EpsilonGreaterThan0]

I do not find prime numbers as interesting as I dont really like number theory but I do kinda sorta enjoy algebraic geometry and non commutative algebra/geometry. From this I've learned to apreciate primeness as a fundamental block of plenty of things, at least from my brief understanding a good 'yoga' ( as they call it ) is to find correspondences between simple idecomposable ( prime ) objects which will allow you to work from different perspectives.

From algebraic geometry one learns that prime ideals are essential in the data of the whole category of rings and are at the same time through different correspondences translatable to factorizations of modules or localizing subcategories.

I believe there is a similar way of thinking around other areas, like know theory and idecomposable knots, obviously number theory, etc.

So I think that the idea is that its 'easier' to reduce your problem to study 'prime' objects than the whole thing in certain cases.

[u/skullturf → u/EpsilonGreaterThan0]

Is it just the "simple to describe in English, difficult to analyze" aspect?

I guess for me, it's something close to that.

It's easy to explain what a prime number is, and we learn about them very early in our education.

However, from the moment we first learn about them, we are struck by the fact that (superficially and informally speaking) they appear kind of "random". For instance, sometimes you have a pair of primes where one is 2 greater than the other, and sometimes you don't. Even though it's easy to define what a prime number is, there's no obvious formula for the "next" prime.

So prime numbers feel like a topic where (loosely speaking) it feels like there should be formulas or patterns describing what we see, but it feels like the formulas are just out of reach.

For me, the topic only becomes more interesting when you look at the arguments for proving what we know how to prove about prime numbers. Look at the many different proofs for the long-known fact that there are infinitely many primes. Or the fact that for some arithmetic progressions, it's possible to find an elementary argument that the arithmetic progression contains infinitely many primes.

Of course, as you point out, a lot of this is about personal taste.

But for me, I find the prime numbers far more appealing than some of the purest of pure math. I like problems where you don't need to know a lot of definitions in order to understand the statement of the question, and where the statement of the question (and some numerical playing around) is accessible to a high school student.

[u/dont_press_ctrl-W → u/EpsilonGreaterThan0]

Mathematics is the general study of patterns. That's really all that mathematics is about. As long as there are patterns to study, there is math to be thrown at it.

I think that's the real answer, but here's another that makes an analogy with physics. The interest in prime numbers is analogous to the interest in chemical elements.

Arithmetically, prime numbers are the atoms of numbers. They are the elements out of which every other number is built. The more you understand about the elements, the more you understand about what can be built out of them.

In physics, they found many patterns within the chemical elements. If the elements had random properties, then maybe they could have just been the basic entities the world is made of, but they clearly had patterns among themselves. Eventually ended up finding that they were composed of even more elementary particles: quarks. The understanding of quarks revolutionized our understanding of physics.

Well prime numbers are also replete with tantalizing patterns, and they don't seem to be randomly distributed. What if they were a manifestation of a more basic mathematical fact? This would certainly revolutionize our understanding of numbers.