related: Nim forum
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This is a basic property of primorials.
You are also making allot of presumptuous statements for which you have no knowledge about.
Anyone who actually reads the paper, and sees the empirical results, AND runs the software, agree it works, even if they don’t understand (completely) why.
I am the expert on Prime Generator Theory (PGT), its math, and applications.
My empirical results are irrefutable. Do you refute them?
Because some people don’t want to learn new things just shows how hard it is for people to change, in the face of irrefutable evidence.
But the advance of knowledge will ultimately lead to widespread acceptance, because it works, and answers heretofore unanswerable questions, and reveals the natural order of the primes and their distribution.
@jzakiya why are you still ignoring this? You’ve obviously written 2-2 which is 0, so the number of twin primes is 0.
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These are primorials. 0# = 1# = 1.
If you really wanted to know about primorials all you have to do is look it up. I’ll post it again.
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but you’ve written (2-2)*(3-2)*(5-2)... which is 0*1*3 which is 0.
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Best of luck. If you ever manage to get it peer-reviewed and published I would be very excited to read & try to understand it better.
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(p_n - 2)# \neq \prod_k^n (p_k - 2). Take n = 3, p_n = 5, (p_n - 2)# = 3# = 6, \prod_k^n(p_k - 2) = (2 - 2)(3 - 2)(5 - 2) = 0.
I hope that wasn’t a central part of your proof.
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I think it actually makes a whole lot of sense that you’ve received the kind of responses you have here, and it’s precisely because the Julia community attracts so many mathematicians.
I only have undergrad experience in math, but when I needed to walk through proofs with professors, my impression was that the job of most mathematicians was less like that of a cartographer and more like a highly skilled hedge trimmer, fastidiously pruning away possibilities until what remains is traceable from start to finish. It is romantic to imagine that these botanists have, in their folly, discarded something valuable, and that you, and you alone, possess the know-how to reconstruct the detritus into a coherent vision. Often, you’ll end up rediscovering the reasons those ideas were discarded in the first place.
The problem is that you have an incomplete link, a conclusion you’ve decided to draw anyway, and a pretty stubborn insistence that empirical evidence and the approval of some folks with (by your own admission) semi-limited understanding of the algorithm is proof enough. It isn’t.
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By the way, there’s another reason I don’t want to implement your algorithm, even if I had the time and skills - it appears you use implementations as endorsements, so by creating a repo I would attach my good name to your algorithm. I’m a physicist, so it’s not that likely, but I don’t want a future employer to google me and find a link to some fringe maths.
Your first step to acceptance must be a rigorous proof, and a peer review of that proof. Maybe reach out to a number theory department with an outline, and they can help you clean it up and get it published, or find a counterexample to refute it. This step will probably be easier if you avoid hubristic gestures like naming your algorithm for yourself.
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What I understand from this thread is that everyone has ignored this, and not even acknowledged it.
Because it’s not an answer to the specific criticism. The paper claims that (p_n - 2)# = \prod_k a_k where a_1 = 0. It simply doesn’t matter that 0# = 1, that product multiplies to 0.
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Is clearly wrong.
Like factorials:
n! = nm=1∏m
Therefor
(n-2)! ≠ ∏ (n - 2)
but
(n-2)! = n-2m=1∏m
For primorials of prime twins:
( pn - 2 )# = ( pn-1 )# ≠ ∏ ( pn - 2)
It seems that this error comes from writing:
pn# = ∏ pn
which means actually
pn# = nm=1∏ pm
so there is no pn on the right side, that you can substitute if you do it on the left.
Don’t know how important this is for the whole picture of what OP does but there seems to be some sloppiness in at least one detail. OP should answer this question in more detail than just referring to some primorial wiki.
I’m not a math guy, but based on the linked wiki, primorials don’t seem to change how subtraction and multiplication works, and a product with the term (2-2) should be 0. The fact that 0# or 1# is 1 doesn’t address why (p_n - 2)# always has a (2-2) term in its product expression.
The definition of primorials in the wiki OP linked also seems to just contradict the way OP uses # ? In the wiki, p_n# denotes the product of the first n primes given the n-th prime number p_n, and n# denotes the product of all primes less than or equal to a natural number n. So either way, it should always be a product of primes. However, you can see two nonprime terms (2-2) and (5-1) in the three lines defining (p_n - i)# in the 4th page of OP’s second manuscript, so OP is by definition not writing primorials.
I’m not a mathematician either but everyone knows that (2-2)=0, surely the OP too. Perhaps the expressions should be interpreted differently. Let’s wait for further explanations from OP. Anyway, some comments above were gratuitously rude.
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I don’t think it’s only a matter of interpretation, OP has responded to the criticism with self-contradicting information. And judging by how heated the thread is getting, especially OP resorting to personal attacks, I’m not sure if it’ll be productive if the discussion continues. Bare minimum for a productive conversation is responding in coherent and polite ways even after hearing “your work is wrong”, and the thread today has not managed that.
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Let’s stop this thread here — we’re far afield from discussing a Julia implementation and are moving in an unproductive direction. Thanks for sharing your work here, @jzakiya.
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