Otherwise you’d end up like the MD who’ve “rediscovered” numerical integration (the trapezoid method) and got it published in the journal of diabetes or whatever.
> Otherwise you’d end up like the MD who’ve “rediscovered” numerical integration (the trapezoid method) and got it published in the journal of diabetes or whatever.
This was shocking because calculus is a required subject in American high schools, and this American doctor presumably went to American high school, not because the doctor didn't check in with mathematicians. Frankly, it would be equally shocking if the doctor had asked a mathematician if integration were a thing, because presumably a doctor is an educated member of society.
Nonetheless, even though it's highly embarrassing to the parties involved, with a little bit of self-deprecating humor if I was the doctor, I could tell people at parties that I, along with Newton and Leibnitz have been published on a foundational numerical integration method. ^_^
It's a blog post, not a peer-reviewed paper. And the tone certainly sounds to me like an invitation to mathematicians "Hey, we're amateurs who found this cool thing, can you tell us more about it?"
I understand what you mean, but how many great breakthroughs in history were where someone shared the foundation of an idea and then another formalized it?
In the more mathematical realm in information theory, turbo codes came out of nowhere by people who were not experts in the field of FEC. Their efficiency far surpassed other methods of the time, to the point that their results in the conference paper were doubted. They didn't have any understanding at the time of why they worked well, they just published results.
Lots of physical phenomena start as simple observations that are later worked into theoretical frameworks. The photoelectric effect comes to mind.
"The initial goal of John Conway was to define an interesting and unpredictable cell automaton....
"While the definitions before Conway's LIFE were proof-oriented, Conway's construction simply aimed at simplicity without a priori aiming at the proof of automaton being alive."
https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
My gut sense is that this has something to do with visualizing the sieve of Eratosthenes, and nothing more than that. But I'd be happy to hear otherwise.
Mostly the "pattern holding up until n^2". I don't know exactly how the groups-of-N factors into all this. Maybe it's nothing and there's no clear relationship.
Yep - we are not claiming to be mathematicians, we're pattern hunters... this is an open invitation to others with more expertise to chime in and help figure this out... it's possible it is a minor discovery or even not a discovery... or it could be useful or even very useful. We don't know. Please help us explore it!
Either way, hearing a mathematician explain why this is a non-discovery would still be interesting to pretty much anyone who doesn't study primes. Kudos on a cool visualization.
If nothing else, this depiction of prime numbers looks very satisfying for recreational mathematics, much more so than the Ulam spiral. I think students who learn how to generate this pattern will be inspired to learn more about math.
”we are not claiming to be mathematicians, we're pattern hunters.”
“Searching for interesting tautologies” or “Hunting for patterns” are good descriptions of what mathematicians do.
Mathematicians do mathematics because they want to be sure that a) they caught a pattern and b) that it is interesting. That’s what’s being discussed here.
These visualization techniques have something going for them. I mean, it's amazing work and with new visualizations, people will make new conclusions based thereon much more easily. Analytic continuation ad nauseum. :)
Is there are particular reasoning or meaning to each dot being 6 numbers? Is there any significant changes if you pick other numbers per dot, following the same pattern?
If the numbers of integers-per-dot was set to 1, the sequence would simply highlight prime numbers as they progress through the pyramid shape since “Parallax Compression” would no longer apply.
Yet they say that this only seems valid for the number of rows equal to the number of integers-per-dot that was used. So, it would indeed highlight with reasonable certainty all the primes in the first row. (I am not a mathematician but I think that is "1" ?)
Assuming you used “and” when you meant “or”, that's trivially true (and redundant) in that all numbers (irrespective of base, which has no effect on this) are integer multiples of 1.
But no primes other than 5 are integer multiples of 5, in any base.
> Senary may be considered interesting in the study of prime numbers, since all primes other than 2 and 3, when expressed in senary, have 1 or 5 as the final digit.
> I expressed it poorly, but not as you state, incorrectly.
No, really, it is completely incorrect to use “multiple of X in base Y” to mean “have X as the final digit in base Y” (which is equivalent to “is congruent to X modulo Y.”)
13 is not, in base 10 (or anywhere else), a multiple of 3.[0]
That's just not what “multiple” means.
[0] Well, the number denoted by the digits “13” in any base that is itself a multiple of 3—other than base 3 itself where “13” is not a valid number—is a multiple of 3, obviously, but we're talking about the number represented by “13” in base 10.
From a cryptography standpoint, could this hint at attack vectors for things like discrete log problems?
I've only learned of the math behind that myself recently, not sure what implications having a "topographical map of the primes" could have, especially if the pattern is relf-repeating regardless of the size of the primes.
> on January 18, 2018, I found a numerical sequence that generated the exact same pattern as Shaun’s pattern
Does this mean that we have a sort of bloom filter-esque test for primality? (ie, it will give you a guaranteed no in O(1) but you'll have to crunch numbers to get the yes?)
If so, are there implications for things that want to know "is it prime?" quickly? Crpytography comes to mind, for instance...
We already have quick algorithms that say "is it prime" with certainty. Reducing the required time from O(log^6(n)) to O(1) isn't particularly important from cryptographic point of view.
Note that log(n) is the length of the prime, so if you are using 2048 bit primes, log⁶(n) is quite large. I don't think anyone actually uses one of the general deterministic primality testing algorithms in cryptographic applications.
we do not have a fast, certain, algorithm for large numbers. The only thing we have for large numbers is “it’s probably a prime. if it’s of a special form, such as messenne, use this other algorithm that takes like 10 days to confirm whether it is actually a prime. If it’s not a number of a special form for which we have specialized algorithms for, which are still pretty slow for large numbers, well you’re SOL, and if you really want to know for certain if it’s a prime or not you better use one of the clever brute force algorithms that do things like only check odds and only check up until the square root of a probable prime, and are still slow as all get out”
So speed improvements are welcome for academic use. Although this doesn’t look like it’s a game changer.
If by "bloom filter-esque", you mean a probabilistic way of testing whether a number is prime or not, then the answer would yes for a majority of numbers.
How? Just run Fermat's little theorem on multiple values of "a" until you feel comfortable. [1]
Why a majority? There are certain exceptions such as the Carmichael numbers to which we need to use slower algorithms to verify the primality of.
Note that I'm assuming the number of calls to an algorithm is constant since it would be naive to discount the size of the number you're testing.
What are the implications of this result? Off the top of my head I can only think of one: the problem of deciding whether a number is prime or not is in the complexity class P (decidable in polynomial time). [2]
How does this affect cryptography? I would say not by that much.
Why? The "hardness" of some forms of cryptography (asymmetric) isn't from the ability to determine primality of a number. It's from the ability of determining the factors of a number. [3] That problem itself has greater implications for the field.
The question of whether it is possible to do so in polynomial time on a classical computer is actually an open question in CS right now.
Note the emphasis on classical! Amazingly, there exist a polynomial time algorithm to do so on a quantum computer called Shor's algorithm. [4]
Just run a search on: prime numbers fractal, and many interesting articles (both academic and other) will surface. Search for: zeta function fractal, and you'll find more. I'm no mathematician but it wouldn't surprise me if two very deep fields turn out to be different "manifestations" of some deeper underlying truth, and I then wonder about the possibility of a larger grand unified theory that includes both maths and physics that would have far reaching insights which would go beyond both mathematics and physics, but then I realise that perhaps I've seen too many movies :-)
I was initially thinking the same, but I don't think that's the case. This will help you find a small range of six potential primes (pick any element on the left edge of the triangle). But you're still required to calculate the factorizations on those six numbers to determine which of them are indeed prime.
No, this won't help with finding primes. As they noted, the pattern for ranges of size k only holds for k lines. So to find a prime of length N (on the order of 2^N), we need to have k=O(2^(N/2)). However, this yields a guarantee that a prime lies in a range of size O(2^(N/2)), which is not particularly useful. We get exponentially better probabilistic results from the prime number theorem.
Just a heads up, but I would pay money for mugs/t-shirts with the pattern.
Of course I can just use to code to make it myself, but it's nice to support people sometimes
Cell i of row j contains n elements of an arithmetic progression, with common difference of j.
If i and j have a GCD != 1, they are not coprime, so they share a factor p, as do all numbers in the sequence, so they cannot be prime
Otherwise it's very likely there's at least one prime
If j is prime every column i is coprime with it, so it's going to be black
The Telegram link leads to a distribution channel, not a group, so one cannot actually discuss it there. I let the author know through the contact feature on the website, as it might be unintentional.
Edit: The link was changed to a group (at the bottom of the post).
Awesome! It's the math enthusiast's dream to come up with something new and exciting outside of academia. Recently I discovered what I thought was an interesting chaotic map, but after posting a question about it to Math StackExchange[1] and emailing one or two professors (no response, which is understandable), my obsession waned and I gave up on trying to figure out if it had any significance. Maybe I should keep trying!
> There's nothing bad about no-response - People are much more likely to respond on the web and email when you're wrong. ;)
There certainly could be something bad about no-response. As someone in academia who gets uninformed musings or crackpot theories from laypeople in his mailbox from time to time, no-response basically means “I know you are wrong, seriously wrong (and, in many crackpot cases, probably mentally ill), but I am not going to just waste my time trying to tell you that.”
As someone else in academia who gets his share of crackpot emails, no response can also mean "oh god, I don't have time to think about this when I have 1000 other emails from my students and colleagues".
(I'm particularly horrid at email.). So I wouldn't take it personally.
I am sceptical of the ability of laymen to even meaningfully understand the field that I am in enough to contribute their own ideas. The amount of literature that would have to be read and assimilated is vast, really achievable only for academics. My prewritten response would only be “Do at least an MA in this field, then we’ll talk”.
Maybe the majority of recipients won't act, or they'll even take offense, but if somebody in there chose to contact you because they look up to your position, you could easily find in a few years that you challenged somebody to get on the right track with one of those responses. Some people are on the fence and looking for a push.
Obviously it's up to you, but I think it's worth a shot.
Who said you couldn't have a hexagonal lattice? Instead of 8, every field has 6 neighbors. The only problem with that really is that you have to expect the 23/3 rule to look differently on that grid. So, no standard glider.
I would also like a clearer explanation of how this was generated, maybe with pseudocode, and would also like to see the larger images mentioned in the post.
As it is, the explanation doesn't really make sense to me.
At GCD(n,k)!=1 all numbers are divisible by GCD(n,k) therefore contains no prime
At GCD(n,k)==1 we have https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith... - so those series contain infinitely many primes - and seems like they actually contain at least one prime in all the pixels of the first N rows (but this should be explained/proved, if it is always true for any chosen N, or just happen to be true for the N-s tried by the OP)
So their picture is nice, and the gcd!=1 cells are all red, but the gcd=1 cells need not be black. For odd N this fails loads of times (always?), and for even N you quickly find failing cells when you start looking for it (see linked comment).
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[ 3.5 ms ] story [ 233 ms ] thread"SQLite looks nice, but I would have double checked with Oracle engineers before announcing this as something novel"
Otherwise you’d end up like the MD who’ve “rediscovered” numerical integration (the trapezoid method) and got it published in the journal of diabetes or whatever.
This was shocking because calculus is a required subject in American high schools, and this American doctor presumably went to American high school, not because the doctor didn't check in with mathematicians. Frankly, it would be equally shocking if the doctor had asked a mathematician if integration were a thing, because presumably a doctor is an educated member of society.
In the more mathematical realm in information theory, turbo codes came out of nowhere by people who were not experts in the field of FEC. Their efficiency far surpassed other methods of the time, to the point that their results in the conference paper were doubted. They didn't have any understanding at the time of why they worked well, they just published results.
Lots of physical phenomena start as simple observations that are later worked into theoretical frameworks. The photoelectric effect comes to mind.
The blog post serves that function.
"While the definitions before Conway's LIFE were proof-oriented, Conway's construction simply aimed at simplicity without a priori aiming at the proof of automaton being alive." https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
“Searching for interesting tautologies” or “Hunting for patterns” are good descriptions of what mathematicians do.
Mathematicians do mathematics because they want to be sure that a) they caught a pattern and b) that it is interesting. That’s what’s being discussed here.
Based on the linked explanation here: https://beta.observablehq.com/@montyxcantsin/unwinding-the-u...
1,2,3,4,5,6
7,8,9,10,11,12
13,14,15,16,17,18
19,20,21,22,23,24
25,26,27,28,29,30
See a pattern? 2 & 3 are the only prime numbers that are an exception. All others are multiples of 1 and 5.
Assuming you used “and” when you meant “or”, that's trivially true (and redundant) in that all numbers (irrespective of base, which has no effect on this) are integer multiples of 1.
But no primes other than 5 are integer multiples of 5, in any base.
https://en.wikipedia.org/wiki/Senary
I expressed it poorly, but not as you state, incorrectly.
No, really, it is completely incorrect to use “multiple of X in base Y” to mean “have X as the final digit in base Y” (which is equivalent to “is congruent to X modulo Y.”)
13 is not, in base 10 (or anywhere else), a multiple of 3.[0]
That's just not what “multiple” means.
[0] Well, the number denoted by the digits “13” in any base that is itself a multiple of 3—other than base 3 itself where “13” is not a valid number—is a multiple of 3, obviously, but we're talking about the number represented by “13” in base 10.
What you seem to mean to claim is that primes > 3 are all congruent to either 1 or 5 modulo 6.
Does this mean that we have a sort of bloom filter-esque test for primality? (ie, it will give you a guaranteed no in O(1) but you'll have to crunch numbers to get the yes?)
If so, are there implications for things that want to know "is it prime?" quickly? Crpytography comes to mind, for instance...
This catches most non-primes ;)
bool maybe_prime(x) { return x % 2 && x % 3 && x % 5 && x % 7; }
https://en.wikipedia.org/wiki/Primality_test#Probabilistic_t...
https://en.wikipedia.org/wiki/Primality_test#Fast_determinis...
So speed improvements are welcome for academic use. Although this doesn’t look like it’s a game changer.
How? Just run Fermat's little theorem on multiple values of "a" until you feel comfortable. [1]
Why a majority? There are certain exceptions such as the Carmichael numbers to which we need to use slower algorithms to verify the primality of.
Note that I'm assuming the number of calls to an algorithm is constant since it would be naive to discount the size of the number you're testing.
What are the implications of this result? Off the top of my head I can only think of one: the problem of deciding whether a number is prime or not is in the complexity class P (decidable in polynomial time). [2]
How does this affect cryptography? I would say not by that much.
Why? The "hardness" of some forms of cryptography (asymmetric) isn't from the ability to determine primality of a number. It's from the ability of determining the factors of a number. [3] That problem itself has greater implications for the field.
The question of whether it is possible to do so in polynomial time on a classical computer is actually an open question in CS right now.
Note the emphasis on classical! Amazingly, there exist a polynomial time algorithm to do so on a quantum computer called Shor's algorithm. [4]
[1] https://en.wikipedia.org/wiki/Fermat_primality_test
[2] https://en.wikipedia.org/wiki/AKS_primality_test
[3] https://en.wikipedia.org/wiki/Integer_factorization
[4] https://en.wikipedia.org/wiki/Shor%27s_algorithm
If so this could be a huge blow to security.
But I must say this is amazing that they were able to visualize prime numbers in this way. These guys are geniuses.
- The right edge of the triangle is always red (ie, no primes present), because it represents (row number) * (3* (row number)+[1..6])
- Prime numbered rows are always black (except the far right column). I can sort of feel why this is true but can't express it mathematically yet.
Cell i of row j contains n elements of an arithmetic progression, with common difference of j. If i and j have a GCD != 1, they are not coprime, so they share a factor p, as do all numbers in the sequence, so they cannot be prime
Otherwise it's very likely there's at least one prime
If j is prime every column i is coprime with it, so it's going to be black
Edit: The link was changed to a group (at the bottom of the post).
[1] https://math.stackexchange.com/questions/2654984/identifying...
Don't forget: There's nothing bad about no-response - People are much more likely to respond on the web and email when you're wrong. ;)
There certainly could be something bad about no-response. As someone in academia who gets uninformed musings or crackpot theories from laypeople in his mailbox from time to time, no-response basically means “I know you are wrong, seriously wrong (and, in many crackpot cases, probably mentally ill), but I am not going to just waste my time trying to tell you that.”
(I'm particularly horrid at email.). So I wouldn't take it personally.
Maybe the majority of recipients won't act, or they'll even take offense, but if somebody in there chose to contact you because they look up to your position, you could easily find in a few years that you challenged somebody to get on the right track with one of those responses. Some people are on the fence and looking for a push.
Obviously it's up to you, but I think it's worth a shot.
Maybe if you shifted everything to the left to line it up, but
As it is, the explanation doesn't really make sense to me.
EDIT: Found this lower down: https://beta.observablehq.com/@montyxcantsin/unwinding-the-u...
Really neat! (I still know I haven't the math chops though. Good place to further my study)
Take the relation to the GCD triangle http://oeis.org/A054521
At GCD(n,k)!=1 all numbers are divisible by GCD(n,k) therefore contains no prime
At GCD(n,k)==1 we have https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith... - so those series contain infinitely many primes - and seems like they actually contain at least one prime in all the pixels of the first N rows (but this should be explained/proved, if it is always true for any chosen N, or just happen to be true for the N-s tried by the OP)
So their picture is nice, and the gcd!=1 cells are all red, but the gcd=1 cells need not be black. For odd N this fails loads of times (always?), and for even N you quickly find failing cells when you start looking for it (see linked comment).