Why is the distribution of primes considered mysterious or unpredictable?
I have posted this in a very few places, but it is met with mostly silence. Presumably it is either such a stupid question that it's not worth responding to, or so profound that everyone who read it had their head explode. Would you do me the favor of explaining what I am missing here? It just seems like the Riemann Hypothesis is a Rube Goldberg detour from what is a pretty straightforward matter. The below is a slightly edited version of what I posted on Reddit. These were two different posts. Both included to hopefully clarify what I am asking.
As long as I know all primes from 2 to n, I can generate the next prime. In fact, in a more messy scenario (because the composites are redundant to the primes), I just need to know the last prime, and I can use all of the previous natural numbers to generate the next prime. This is all rather mechanical. Yes, it will take some calculating, and the computer will eventually slow to a crawl and run out of resources if you go large enough, but it's basically gears meshing together that could be made into a machine c.1800's or earlier. It seems that the Riemann zeta function is a very roundabout means to show the distribution and is no less calculation intensive. Clearly, I am missing the point of pursuing a proof of the RH. Any clarifications?
Many of the online explanations of the RH stop at just stating that it tells us "something" about the distribution of the primes. Others go as far as generating the waves from the zeros and summing them to reveal spikes or steps at the primes. Cool stuff, but is that all there is to it? If so, then I'm guessing that there is a way to use the Riemann zeta function to describe the functioning of my 1800s pile of gear-based counters. They are both spitting out prime numbers. However, my imaginary* gears (or my really crude brute-force Python code) do it with much less opacity. Most likely, I am missing some fundamental point here. Just looking for someone to explain what that point is.
And a final note....The above is discussing just the base primes (powers of 1). However, the higher powers are also generated in a mechanical manner...no mystery.
*Not a reference to SQRT(-1)
I attached the python code for "mechanically" generating primes. This is to illustrate the point that it can be done. It is kept simple to make the point clear, so optimizations are not present.
Thank you,
Paul Berry
Pberry
Answer
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Kav10
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Wow, so you've taken a minute to think about it I see. It's a great answer and much appreciated. More than I expected. Problem is that I have more questions based on your answer. Would you be willing to iterate? I am willing to pay again.
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Sure. Which part of it do you want to know more about, or which parts caused more questions for yo?
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Should we do this here in this message box? Very limited characters. But a few points: Not sure I understand the distinction between the two, but if by "how" you mean what is the mechanism, then I think the gears illustrate how/why much better than the zeta. Gears having to align just so for a prime to appear, and that becomming less frequent as more gears
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are added, and why it is both basically predictable by the prime counting function, and yet occurs with jaggedness. The gears may not be the reality, but they illustrate it perfectly/accurately. It may just be that we have to figure out that predictive formula, but laid out before us is a machine that does exactly the "same thing" as the reality. Also, it seems that calculus get to cheat a bit by using every natural number on the planet, but I don't get to iterate infinitely. So the ground
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rules on what constitutes a valid solution method are unclear to me. Also does success look like a formula that can predict just the next prime, or does it have to be able to predict any arbitrary prime? Sorry, I was supposed to quote this for my first question: In summary, I think the gears give us a mathematically crackable model where we can actually see what is happening.
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The RH gives us "one of the most difficult unsolved problems in mathematics". The editing capability here (and the quantity of mistakes that I make) is not great. Just above, I tried to quote something and it somehow did not appear. Here it is again....the reference statement....the basis for my first question... "Because the Riemann zeta function encodes information about how primes are distributed, not just where they are."
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Yes, it is hard to interact here. You can ask a direct question. Click on my profile, and you see the option "ask direct question". I'll answer your questions here, for now.
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The key is what kind of “how” you are answering and there are two fundamentally different “hows”! Mechanistic "how" and Statistical/Global "How". First one explains each prime as a consequence of divisibility logic but it does not tell you anything about patterns among all primes. It only gives you the next one after lots of grinding. It is local, sequential, and deterministic. It is what computers do (humans historically did).
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The second one is a global, bird’s-eye view of primes! The zeta function enters here because it encodes this global information. Unlike answering the previous "how", this approach does not need to compute each prime and estimates their density and structure across ranges. Think of it as earthquales vs. seismology. You have a machine that simulates earthquakes. But now you want to understand the frequency and size of earthquakes worldwide. So, you build a statistical model ...
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Gears vs. Zeta are basically two views of the same landscape. The gear model does not give you a formula for the distribution, just a way to compute each prime in sequence. The gear model shows how primes are filtered. The zeta model shows how regularly primes are spaced on average.
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Your model builds up reality one number at a time. Analysis takes an infinite sum/product and manipulates it like a single object. Your model is more grounded, but it does not give you that “big picture” shortcut.
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You are not misunderstanding RH. You are approaching it from a constructive, algorithmic mindset. An engineer-minded approach. I am also an engineer. Your gear and RH are not in conflict. RH does not replace your model. It just answers a different, more abstract question that we cannot solve with brute force alone.
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Maybe you have basically found the "fault" line between computation vs. estimation ;-)
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So if I can figure out how to mathematize an infinite series of gears into a function, then that would be valid? Iteration is not essential, it was just a way to filter out the composites. The composites are just redundant to the primes in a gear counter context. For instance, the "6 gear" is only at modulus when both the "2 gear" and the "3 gear" are at modulus. Maybe the noise from the composites would appear similar to the non-prime peaks (prime powers > 1) in the Riemann harmonics plot.
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Yes. We are touching the surface of a deep idea! If you can mathematize your gear model into a function that captures the essence of it and compressing all the gear logic into something analytic. And that would absolutely be a valid and valuable formulation. The primes generate all the structure, and the composites are just reverberations of that structure. This discussion can continue on and on to get to Fourier analysis, Euler's Product, ... You are reverse-engineering a universe ...
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I can help you with your Python code (I am more proficent with R, but Python is fine), if you want. Submit a direct question for that.
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There is no attachment!
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Ok, there should be a .zip attached now.
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I’ll take a look tomorrow.
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Your code is interesting! Here are a few comments/ideas: You could store and return the list of primes if you want to compare against known distributions. You could also use this to test hypotheses about prime gaps or compare against expectations from the Prime Number Theorem. Maybe track and plot which gears fire at each number. and it might reveal visible wave interference patterns? (similar to how Riemann harmonics stack up).
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Compare against the Möbius Function μ(n)? I see your gear system as a practical way to simulate μ(n) values up to a point. Compare against primepi(x) or even plotting both to sort of validate or see if you can find a growth trend? sympy.primepi(x)
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I think we may have broken the commenting feature (overflowed?). I tried to add 497 characters and it turned red and said the comment is missing. If this posts, I'll try some smaller posts.
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I once had a pretty neat (meaning that even I could do some cool things with it) signal processing software (Sigview). Maybe I'll buy the latest version and look at what appears. There's a lot to look at here. I'll investigate those functions and see if anything clicks. I would have to optimize the code and write it in a compiled language for it to be useful for larger numbers. However, it will always bog down at some point so I don't know how much comparison/testing it will be useful for.
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Yes, there’s a limit on number of characters, make sure you have less than limit before hitting submit!
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Back to what I mentioned before, you can ideally perform a Fourier transform on the binary primality signal and look for frequency components that dominate. Those might match known prime gaps, twin primes, or echo zeta harmonic structures. Sounds very interesting!
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At the very least, you can validate known behaviors such as average gaps, harmonic spikes, etc. and then test enhancements!
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So, your formula detects whether a number is prime. What you don’t have and what makes prime distribution “mysterious” is any known formula or closed-form expression that predicts the n-th prime directly and/or predicts where the next prime is, without searching. It also does not describe the exact spacing between primes.
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So even though we can see exactly why the distribution would be jagged and on average wider as the numbers grow, we only win if we can say what each gap is out to infinity? Have you ever considered lowering your standards? ;-) Conceptually, I have an idea of how to get a function to produce any arbitrary prime. However I don't know if I can make it completely non-algorithmic. I don't know what is legal in an infinite series. I'll try to flesh that out.
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That might be the most elegant summary of the entire Riemann Hypothesis problem I have heard ;-) And yes, you nailed a deep truth here, you do have good explanations for why the distribution of primes gets jagged and sparse (your gear model makes that visual and mechanical), but the mathematical world still demands an explicit, general formula for exactly when and where primes occur (out to infinity) without brute force!
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The bar is so high because it is the difference between understanding a system and controlling it. If you have a conceptual sketch of a function that gives you any prime at position n, that is worth pursuing. It is a million dollar question!
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If your function uses infinite terms but converges quickly and deterministically, you might still be in the game.
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And now I try to split a hair.... "So, your formula detects whether a number is prime. " I like "detects" over "tests", but I don't know if that is fully correct either. I think it says "This number is prime, because it can't be anything else." It declares a number to be prime when there are no other primes available to compose the current number. Is that an important distinction? I don't know.
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You are not just splitting a hair, you are touching the philosophical and mathematical core of primality! And yes, the distinction you are trying to draw is important.
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Testing is external and agnostic, it presumes nothing about the number until evidence is gathered. Detecting is still observational, but more system-internal! Your gear model fits this nicely, it detects a number that does not align with any previous gear. It feels more like saying this system is monitoring numbers and alerts when a condition is met (no smaller primes divide this one).
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On a more philosophical note, I wonder if mathematics is trying to accurately reconstruct a novel by reading only the CliffsNotes. We're talking whole numbers, but that is less than a nanoscopic subset of all numbers. It might be that the answer to "why" the primes are this way, is because this is the only solution for the very limited question we asked. Not sure how this helps at all.
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It does help, because you’re asking whether we are mistaking the surface complexity of a narrow problem for deep mystery.! That is foundational. Either way, your gear model and your perspective is asking that bigger question and that is the best place to be!
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What your system does is not mere detection or testing. I mean it more fits detecting, but it is constructive realization of a number’s irreducibility within the evolving structure of the naturals.
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It is not asking if something is prime. It is basically saying you have no ancestors in this system, so … you are now one of the founders!!
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If we keep splitting hair, we would easily spend hours on this.
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Well, now I have something to think about on the 3-hour drive home today. Thank you for your time and serious engagement in this discussion.
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No problem. That’s a long drive! Safe travels!
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If you have more questions, please submit a direct question so things don’t get too messy in comments!
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You are asking about the RH, which is not my area of expertise, so I won't take the bounty. But I want to clarify that we don't know what the NEXT prime number is, given a list of prime numbers up to n. We know there's a prime number after it (and we can construct it), but it's very likely that we are missing primes in between.
Thank you for that quick and relevant response. Maybe semantics is one of the issues here. I can calculate the next prime number. Perhaps not in a way that qualifies as "knowing" what it is. I do not know the rules for a proof. If I start from that last prime and keep incrementing, I know that I have reached the next prime when I cannot construct that number from the previous primes. There is a mechanical means to know when that point is reached. I do have to test each number up to then.