A brief explanation of why primes peak at repeated multiples from a layman who's wondered why before. Obvious first example all primes above 2 are of the form 2x+1. An obvious repeating pattern of primes. You can take this a small step further. All primes above 6 are of the form 6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of numbers are worthy of being considered prime. This is a slightly le…
This pattern forms the basis for wheel factorization [1], a faster way to factor a number than naïve trial division. [1] https://en.wikipedia.org/wiki/Wheel_factorization
Surprising hidden order unites prime numbers and crystal-like materials
31–40 of 41 posts
Re: Surprising hidden order unites prime numbers and crystal-like materials
#32Earlier quoted context omitted.
I've heard this sentiment before and I don't really understand it. There's no separating physics and math. Keeping math "pure" really means keeping it "purely abstract", so it resists any kind of practical application.
> There's no separating physics and math. There is. Even though most physics research is extremely mathematical and abstract these days, it's still ostensibly grounded in empirical science. Math is not science, it just provides useful tools and insights for studying science. Unlike physics, the disciplinary imperative of math is not to provide us with truths about this world or any other world. Its imperative is to t…
While I certainly understand that math and physics are separate concepts, physics as we know it wouldn't be possible without math. I'm sure in your mind you can separate them, but if you took math away from physics, we wouldn't have modern physics.
Math is how physics is given practical application, if anything, that means science is more abstract than math is.
Re: Surprising hidden order unites prime numbers and crystal-like materials
#33A brief explanation of why primes peak at repeated multiples from a layman who's wondered why before. Obvious first example all primes above 2 are of the form 2x+1. An obvious repeating pattern of primes. You can take this a small step further. All primes above 6 are of the form 6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of numbers are worthy of being considered prime. This is a slightly le…
In your example you started at 1/3 (33.33%) and moved to 58/210 (27.62%). What does this limit approach ad infinitum?
Re: Surprising hidden order unites prime numbers and crystal-like materials
#34Earlier quoted context omitted.
This pattern forms the basis for wheel factorization [1], a faster way to factor a number than naïve trial division. [1] https://en.wikipedia.org/wiki/Wheel_factorization
Also, the https://en.wikipedia.org/wiki/Sieve_of_Atkin
Author of this thread here. That article states that the above sieve was created in 2003.
I actually wrote and described it back in 2002 - https://forums.overclockers.com.au/threads/show-off.71630/#p...
I didn't think much of it back when i did it and i'm not a mathematician, just a hobbyist. I guess i should publish my prime hobbies more...
Re: Surprising hidden order unites prime numbers and crystal-like materials
#35A brief explanation of why primes peak at repeated multiples from a layman who's wondered why before. Obvious first example all primes above 2 are of the form 2x+1. An obvious repeating pattern of primes. You can take this a small step further. All primes above 6 are of the form 6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of numbers are worthy of being considered prime. This is a slightly le…
In your example you started at 1/3 (33.33%) and moved to 58/210 (27.62%). What does this limit approach ad infinitum?
You can have a look at https://en.wikipedia.org/wiki/Prime_number_theorem
Re: Surprising hidden order unites prime numbers and crystal-like materials
#36I read once that prime numbers were a key element element on cryptography (because it's easy to multiply two prime numbers, but difficult to say if a number is a multiple of two prime numbers, if I remember correctly). Will this discovery have negative impact on it?
Yes. If we can quickly generate large primes in a sequence then we can find the products easier.
That’s not really true. We don’t find new primes to make cryptography safe.
We already have very large lists of primes. It’s choosing which primes are used that is the expensive part of attacking crypto.
Additionally the OP simplified how cryptography works too much in saying you just need to find the products to decrypt.
Re: Surprising hidden order unites prime numbers and crystal-like materials
#37Earlier quoted context omitted.
It doesn't matter to the algorithm that we know if it's a product of two primes. Anyone attacking an RSA private key knows that it is. The only unknown is which two primes.
Which are easier to generate with a prime formula..
This is wrong. We have lists of all primes used in crypto already. Just knowing them all doesn’t make attacks quicker.
Re: Surprising hidden order unites prime numbers and crystal-like materials
#38Earlier quoted context omitted.
I don't agree with this. And honestly, I really think that depends on what foundation you rely on to think with, work with, create with, test with, and check your own tests with. Physics does not have to use itself to understand itself. Math does.
> Physics does not have to use itself to understand itself. Could you explain what you mean by that?
It's simple. Physics uses mathematics to construct equations that describe properties of physics.
The difference is physics has reality to test against - observations that can be measured. Math does not have this. Math's only metric against itself is itself.
Re: Surprising hidden order unites prime numbers and crystal-like materials
#39Earlier quoted context omitted.
> There's no separating physics and math. There is. Even though most physics research is extremely mathematical and abstract these days, it's still ostensibly grounded in empirical science. Math is not science, it just provides useful tools and insights for studying science. Unlike physics, the disciplinary imperative of math is not to provide us with truths about this world or any other world. Its imperative is to t…
While I certainly understand that math and physics are separate concepts, physics as we know it wouldn't be possible without math. I'm sure in your mind you can separate them, but if you took math away from physics, we wouldn't have modern physics. Math is how physics is given practical application, if anything, that means science is more abstract than math is.
It's about direction of influence.
Physics should not influence math. Math has to be a consequence of it's own axioms, or have it's properties tested against it's own constructions.
Otherwise math falls apart. Then it's useless for physics.
Practical application is wonderful for math. But the direction math is crafted and interpreted matters a lot.
Re: Surprising hidden order unites prime numbers and crystal-like materials
#40Earlier quoted context omitted.
Which are easier to generate with a prime formula..
Define "easier" in this context. If you have an algorithm whose complexity increases exponentially with the key length, saying you made a 10% gain doesn't mean squat.
And it's wrong already. We already know all primes uses in encryption (just download a prime-number list).