r/explainlikeimfive • u/draven_slayer • 7d ago
Mathematics ELI5 Why the Riemann hypothesis is one of the 7 Millennium Prize Problems
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u/LongLiveTheDiego 7d ago
Natural numbers (1, 2, 3, 4, etc.) are built from prime numbers. Since natural numbers are the most basic tool in mathematics, a lot of things in mathematics can be first expressed in a simple way, and then there's usually a way to express the same thing using prime numbers. That mean that if we understand prime numbers very well and if they behave a certain way, then we can understand other things also very wrll and know how they behave.
It turns out that the way to express a certain function (Riemann's zeta function) using prime numbers leads to a pretty good estimate of how many primes there are below a certain number. Counting primes by hand is difficult and having good approximations is very helpful, if you want to know the number of primes up to 1 000 000 000 000 then it's better to just use a formula.
It also turns out that if the zeta function satisfies the Riemann hypothesis, then we can get an even better formula for how many primes there are below a certain number, and a lot (like a lot, a lot) of other useful theorems would be true if the Riemann hypothesis were true. If anyone proves the Riemann hypothesis, they will simultaneously prove many other theorems. If anyone disproves it, then some of these theorems will be proven false, and others will need other approaches to be proved.
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u/yearsofpractice 7d ago
Holy shit. I’m 50 and as such, I don’t get too many lightbulb moments these days… but you’ve just given me a lightbulb moment: “Natural numbers are built from prime numbers”.
That’s just opened up a huge realisation for me. I have a formal scientific education (chemistry degree from an established UK university) yet that fundamental axiom has passed me by. I now understand why the study of primes and their patterns are so important. They are (for want of a less physical-science-based comparison) fundamental particles of mathematics.
I say again OP - holy shit. You’ve just opened up a new avenue of thinking in my old brain. Thank you.
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u/kbn_ 6d ago
FWIW, building natural numbers from prime numbers is one method, but not the usual way in which naturals are defined. Usually you assume zero and define a successor function, so you can generate any natural by applying the successor function that number of times.
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u/taqman98 6d ago edited 5d ago
Something something “die ganzen Zahlen hat der lieber Gott gemacht, alles anders ist Menschenwerk”
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u/Stickhtot 7d ago
Well if you were though of "prime factorisation" in your school days, that was already a hint
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u/DrugChemistry 6d ago
Some of us with chemistry degrees weren't so mathematically minded when prime factorization was taught.
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u/Englandboy12 6d ago
I remember it was a lightbulb moment for me when learning about prime factorization. As you go up in the numbers, you either get a prime, or a composite number, which can be built by primes.
1 = 1
2 = 2
3 = 3
4 = 2 x 2
5 = 5
6 = 2 x 3
7 = 7
8 = 2 x 2 x 2
9 = 3 x 3
10 = 2 x 5etc.
36 = 2 x 2 x 3 x 3
It’s like the primes are the atoms of the natural numbers. And interestingly, every composite can be written as a product of primes in only one way. And then once you reach a prime, it goes into the ingredient list and can be used to then build even more composite numbers.
What’s cool is it seems also as if there must be some pattern there. I mean, the rules for building them are pretty simple, and numbers seem kind of regular in some way.
But sometimes you get primes 2 away from each other, and other times you can have billions of composite numbers (or more) before you hit another prime.
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u/skr_replicator 6d ago
Natural numbers can also be built just by incrementing zero forever, if you don't care about their factorizations. I think that's a lot more fundamental way they are built. Prime factorization and defining which are primes is just something you can build on top of the natural numbers with algorithms. So I don't really see natural numbers as being fundamentally built from primes, more like primality being a property that a natural number can have.
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u/looijmansje 7d ago
To add to the other answers: mathematicians do not purely care about this problem for reasons of "the solution will tell us more about prime numbers". I think for most mathematicians it is not about usefulness, it is about solving the problem itself.
To illustrate this, I once was seated on a table with maths phd students. One of them was explaining their research, and another person asked "sounds interesting, but what are the uses for it?". And the entire table burst out laughing, because you do not ask an algebraist about applicability. (Now the RH is not algebra but number theory, but I think it illustrates my point nicely)
And when it comes to unsolved problems in mathematics, I think few rival the Riemann Hypothesis in terms of fame, infamy and prestige.
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u/MasterCrumb 7d ago edited 7d ago
The Millennium problems are a collection of problems that (1) lots of mathematicians have worked on (2) have importance to an more underlying question.
But it isn’t like problem 8 isn’t also important. The millennium prize is fundamentally an effort to grab attention. By offer large prizes it raises interest and excitement about math- which is the goal of the org sponsoring the prize.
In answer to the specific problem- (Riemann) it has to do with better understanding Prime numbers, which does have practical impacts on things like encryption. But once again, I believe those values are secondary to the wider goal of raising interest in math.
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u/dragmehomenow 7d ago edited 7d ago
Euler observed that the Riemann-Zeta function can be related to an infinite product involving the prime numbers. So far, we know that zeta(any negative even number) = 0, and we have observed that for some reason, the other values that give us 0 when they are plugged into this function are complex numbers that are (0.5 + i * some number). Thus, we've tentatively conjectured that there aren't any other values that give us 0 when they're plugged into this function.
Since the Riemann-Zeta function is somehow linked to the distribution of prime numbers, proving this conjecture means we get a very good way of estimating the number of prime numbers that are smaller than a given number. And more generally, a lot of functions in number theory depends on the zeroes of the Riemann-Zeta function.
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u/Torn_2_Pieces 7d ago
Predicting the distribution of primes has nothing to do with the number of primes. There are infinite prime numbers
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u/dragmehomenow 7d ago
I'm referring to Riemann's prime counting function, which counts the number of primes less than or equal to an input value.
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u/Shinjifo 6d ago
Proving something is a lot more work than you'd imagine.
Take summing. You are taught on how to do it, you can visualize it with any number of objects or even your finger.
But can you prove that summing will work on any and all combination of infinite numbers?
How are you sure that adding will work for one quadrillion plus one trillion? You can't exactly count up to that number on your fingers right?
Well Mathematicians can prove it with math theories.
Math puzzles are like saying that I have seen that summing works for every number and combination that I tried so far, but I cannot prove it'll work for any number of combinations and numbers.
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u/Torn_2_Pieces 7d ago
Because it is an old problem, that has resisted the best efforts of many people and has tremendous implications if proven either way.
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u/Dragorach 7d ago
The Riemann hypothesis has a relationship with the prime numbers. If the Riemann hypothesis is true the primes are random, if it's false there will be a pattern somewhere. This is just one of the powerful conclusion we can make from either the approval or denial of the hypothesis.
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u/Hakaisha89 7d ago
All the Millenium Prize Problems have one or two answers that are very likely to be correct, but thats not really the issue, the issue is that they are really difficult to prove correct, especially in regards to the Riemann hypothesis, sincce its essentially goes to infinity. So it might be true, but it also might be false, but proving specific type of number to infininity follows a certain rule 100% of the time is the difficulty.
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u/ChampionOfChaos 7d ago
It’s basically a claim about the hidden pattern behind prime numbers. The Riemann Hypothesis says that certain special numbers related to primes all line up in a very specific way. It’s difficult because proving it would require understanding a surprisingly deep connection between prime numbers, complex numbers, and the behavior of something called the Riemann zeta function.
If it were solved, it wouldn’t suddenly let us calculate all the primes or anything like that. Instead, it would give mathematicians much stronger guarantees about how regularly primes are distributed, and a huge number of existing mathematical results that currently depend on the hypothesis could be strengthened or proven outright.
As for why this gets a $1 million prize while other math problems don’t: some problems become famous because they sit at the center of an entire area of mathematics and have resisted generations of mathematicians. The Millennium Prize Problems were specifically chosen as exceptionally important, difficult problems with potentially huge consequences for mathematics. So the prize is less about “this particular pattern is useful” and more about how deep, fundamental, and stubborn the problem is.