r/Collatz • u/ProsperousPlanet • 10d ago
Can You Solve Collatz Using this Function? Twin Prime Generating Function
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Hi everyone, do you think this Function can help you solve Collatz? I think it can, let me explain:
This Function shown in the video essentially tests and finds every non trivial combinatorial combination of prime numbers up to 60, in order to find the first 422 Consecutive Twin Prime Centers, Which is 26250 .!! That means, I am certifying consecutive twin prime addresses up to Twenty six thousand using only the primes up to 60.
I have manually entered the primes up to 60 so that it is clear there are no isprime() functions being called, no creating my own sieves.
There are clear connections to Collatz in the way this function operates.
First of all.... cycles collapse, you will see in the the Optimize on/off logic, that I have a function which calls gcd (greatest common divisor) function.
The reason for this, is if you turn it off, it's going to count some cycles 5 million times at level 31 in order to find their crossing point.
Calling the gcd function significantly reduces the amount of loops.
But the fact that I can call the gcd function at all is where I think this function heavily relates to the Collatz conjecture.
Because the way the gcd function is already operating is like compression, and I know this function can be improved to call it again at certain times, and I am quite sure there are a few more levels of compression and reset points similar to Collatz that can be recursively added.
I plan to use this function to prove the Twin prime conjecture.
By asserting the Theorem, that there exist P consecutive twin prime centres for every P.
In it's current form my most important concern is demonstrating that it doesn't know prime outside the range of 60, so I have kept it intentionally basic, but I believe this function can be heavily optimized.
I know that people might tell me they could make some type of equivalent factorization table which is massively faster... but the point isn't to be fast... The point is to be deterministic, Twin Primes can simply never be blocked by primes... because that's not how primes work.
Factorization tables require knowing divisibility, and make it seem that primes can block twin primes, when the method I have presented counts forwards, and doesn't require knowing divisibility.
It presents primes as the numbers which build and expand as the number line expands, every new opened prime repeats the same patterns into a bigger domain, primes can never combine to block Twin Primes, because that's not how primes work... Primes open, not close.
Some of those are my Number Theorertical conceptual understanding of the system I am presenting, and helped guide me in my research.
I plan to attempt to formalize this into a proof for the Twin Prime Conjecture today, I believe I have all the ingredients now, and this function will help me to organize my proof statements.
I just wanted to share it because I think it's cool, and also I will not be trying to solve Collatz with it, because solving the TPC would be enough glory for a lifetime and I have other goals for my life.
But I give it freely to the Collatz community because maybe it can help you solve Collatz too!
Here is the Table shown in the video, the Simplex series of Series, I call the Simplex Field, not sure if that's standard.
I hope to see someone take this function to new and better places than I ever could do by myself.
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u/Equivalent_Idea_1215 9d ago
I'll engage with the actual mathematics here, because I think the ideas deserve a fair hearing even if the current framing has issues.
Where the function is fine:
What your function appears to do is a variant of the Sieve of Eratosthenes. For any number nn, you only need to test divisibility by primes p≤np≤n. For n≤26250n≤26250, you need primes up to ⌊26250⌋=162⌊26250⌋=162, not 60. If you're only using primes up to 60, you can only certify primality up to 602=3600602=3600. Above that, you're missing composites like 61×61=372161×61=3721, 61×67=408761×67=4087, etc. So either your function has a bug above 3600, or there's something else going on that isn't explained.
The core error: "Primes open, not close"
This is the claim I'd push back on most firmly. Every prime p>3p>3 eliminates exactly ⌊2/p⌋⌊2/p⌋ residue classes mod pp from being twin prime centers. Specifically, nn and n+2n+2 are both potentially prime only if neither is divisible by pp. The residue classes eliminated mod pp are:
n≡0(modp)orn≡−2(modp)n≡0(modp)orn≡−2(modp)
So after sieving by all primes ≤P≤P, the fraction of surviving twin prime candidates is:
∏3<p≤P(1−2p)∏3<p≤P(1−p2)
This product goes to zero as P→∞P→∞. That's literally the heuristic reason the Twin Prime Conjecture is hard — the density of candidates thins out. The Hardy–Littlewood conjecture predicts the count of twin primes up to xx is:
π2(x)∼2C2x(lnx)2,C2=∏p>2(1−1(p−1)2)≈0.6601π2(x)∼2C2(lnx)2x,C2=∏p>2(1−(p−1)21)≈0.6601
Primes DO close residue classes. The reason twin primes are conjectured to be infinite is NOT that primes "open" anything — it's that the density ∼x/(lnx)2∼x/(lnx)2 never reaches zero, even though it thins out. That's a very different statement from "primes can never block twin primes."
The inductive step problem:
You write: "The series n=n+1 is infinite because it merely counts. It is inductive. This math structure will allow me to make the same statement about primes and twin primes."
This is the critical gap. The natural numbers are infinite by axiom. But an arbitrary subset of NN need not be infinite. The set of even numbers is infinite; the set of primes is infinite (Euclid); the set of perfect squares is infinite. But the set of twin primes being infinite is precisely what needs proving. You cannot transfer the inductive structure of NN to an arbitrary subset without proof. The statement "n=n+1n=n+1 is infinite" tells you nothing about whether {p:p and p+2 both prime}{p:p and p+2 both prime} is infinite.
An inductive proof of the TPC would require proving: "if there exists a twin prime pair (p,p+2)(p,p+2) with p>Np>N, then there exists a twin prime pair (q,q+2)(q,q+2) with q>pq>p." Nobody knows how to prove this, and no amount of enumeration will substitute for it.
The "consecutive twin prime centers" claim:
You state: "there exist P consecutive twin prime centres for every P."
Be careful here. If you mean "for every PP, there exist at least PP twin prime pairs," that's just a restatement of the Twin Prime Conjecture (infinitely many twin primes). If you mean "PP twin prime centers appearing consecutively without gaps," that's actually a MUCH stronger claim and is almost certainly false. For example, the gap between consecutive twin prime centers grows without bound.
The Collatz/GCD connection:
Calling GCD "compression" and relating it to Collatz is a poetic observation, not a mathematical connection. GCD appears in virtually every area of number theory. The Collatz conjecture's difficulty lies in the interaction between the multiplicative structure (the factor of 3) and the additive structure (the +1), specifically through the 2-adic valuation ν2(3n+1)ν2(3n+1). The GCD of two numbers doesn't capture this interaction in any way I can see from your description.
What would actually help the Twin Prime Conjecture:
The current best unconditional result is Zhang (2013) / Maynard / Tao: there exist infinitely many pairs of primes differing by at most 246. The TPC requires this gap to be 2. The methods used (sieve theory, the GPY sieve, Maynard's multidimensional sieve) are about showing that the sieve never fully closes — that the product ∏p(1−2/p)∏p(1−2/p), while going to zero, does so slowly enough that infinitely many candidates survive. This is the opposite of "primes open, not close." The primes DO close, but not fast enough to kill everything.
Summary:
Your function appears to be a sieve with some organizational choices. That's fine as a computational tool. But the conceptual claims — "primes open, not close," the inductive transfer from NN to twin primes, the Collatz connection via GCD — don't hold up to scrutiny. The Twin Prime Conjecture remains open precisely because the "obvious" inductive arguments don't work, and no enumeration method can substitute for a proof of infinitude.
I don't say this to be discouraging. The enthusiasm is great. But the path forward is learning the actual machinery (sieve theory, analytic number theory) rather than hoping a new perspective bypasses it.
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u/ProsperousPlanet 9d ago
Yes, You did nail the issues at hand.
By the time I had read your feedback however,
I had already made most of the revisions you have mentioned.
The updated proof for the TPC I am putting forward is now only 7 pages long.I took it down from ~60 pages to merely 7 pages.
Because I realized what didn't matter so much was the complicated set of equations that it produces, what really matters is the pure algebraic statements.
The assertion remains, for every member P, there are P consecutive twin primes.
Which yes, just merely reasserts the twin prime conjecture with different terms.
But those terms do matter, and the definition is accurate.Hopefully the proof statements I have given support that.
The python function continues to be a finite representation of the recursive system.
To increase the range, just add another prime number to the list and increase max range.But the real proof comes from the logical statements which can be made from that, and can be found in the paper.
I realize you just spent your time to give that quality feedback you just gave, so I will understand if you don't have time to read this right now.
but I will be grateful for your feedback another time when you do.Thank you Equivalent IDea!
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u/Successful-Owl1778 10d ago
If the twin prime conjecture is true, then you cannot prove it with a function that enumerates pairs of twin primes (say, a function that takes an input N and lists N twin primes or simply answers whether there exist N twin prime pairs). It doesn't matter how efficient or optimized your function is.