71–80 of 141 posts
Earlier quoted context omitted.
You can also approximate pi as 22/7, which explains some of the other patterns that appear farther out.
> You can also approximate pi as 22/7 Except in Indiana, where the legal value of pi is 3.2 by mandate.
One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
> imperceptible isn't that sort of cheating, when it comes to math? It's "almost" the thing just not the thing
You just have to properly define what you mean by "imperceptible" and you are good to go. Now you can look for a bounded approximation. See how it changes as things get bigger. See if it can be improved.
Bounds and approximation are generally a very useful tool. Properties simplifications and working on sub parts can also yield interesting results.
Generally I would say that if you can’t prove something, trying something close but simpler is nearly always a good idea.
One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
Yes, it was such a thrill to discover this in college (not a math major). I formulated it as all primes >2 are “factors” of 1 and 5 in a base 6 notation.
And yes, in base 6 that becomes 1 and 5, as p%6 = 1 or 5, equivalent to p=6k±1. Interesting observation, thanks :)
Earlier quoted context omitted.
Hence the video what? It's polar coordinates making the spiral, not prime numbers.
The primes >3 are all of the form 6k+1 or 6k-1, so they are close to 6k. As k increases, the resulting marks on the polar coordinate system form nearly complete turns because 6 is close to 2 pi. Hence, spirals. See my other comment for a clearer explanation of why primes >3 all must necessarily be one more or one less than a factor of 6k where k is an integer. That is the crucial piece that the author did not make to…
Bit of a let down when they reveal that plotting integers in general results in the same spiral pattern as primes. So naturally primes as a subset of integers also produce spirals. Seems the title is a bit misleading!
I found it insightful, not a let down. It says a lot about human nature and our biases towards spotting patterns that may be incomplete or inaccurate.
One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me. The reasoning, which is in the article here, is that you can make any whole number you wish if the number is…
Wouldn't the vast majority of those studying primes learn this from their textbook?
Note that 6k-1 is the same as 6k + 5. By writing that way, we can focus in positive representations of the modulo 6 congruence.
6k + 0 can't be prime, it's divisible by 6, yielding k
6k + 1 might be prime: we cannot rule it out by division.
6k + 2 cannot be prime, it's divisible by 2, yielding 3k + 1.
6k + 3 cannot be prime, it's divisible by 3, yielding 2k + 1
6k + 4 cannot be prime, it's divisible by 2, yielding 3k + 2
6k + 5 might be prime again.
That covers all cases of the modulo 6 congruence.
Thus only 6k + 1 and 6k + 5 can possibly be prime.
This is trivial fluff, only a smidgeon more clever than "all primes greater than 2 are of the form 2k + 1".
Another line of reasoning:
If a number N is divisible by 6, then it is even. This means that N + 2 and N + 4 are also even. Thus none of those numbers are prime.
If a number N is divisible by 6, it is also divisible by 3. This means that N + 3 is also divisible by 3. Thus, it cannot be prime.
That leaves N + 1 and N + 5, whose divisibility doesn't relate to 6.
I can also make all integers look like sine waves sin(x)
Plotting primes p like this is plotting p*e^(ip), but the spirals are an interesting observation. You can straighten them out by adding a pi factor to the polar coordinate (so iside the sin...) but that's less interesting.
Earlier quoted context omitted.
The primes >3 are all of the form 6k+1 or 6k-1, so they are close to 6k. As k increases, the resulting marks on the polar coordinate system form nearly complete turns because 6 is close to 2 pi. Hence, spirals. See my other comment for a clearer explanation of why primes >3 all must necessarily be one more or one less than a factor of 6k where k is an integer. That is the crucial piece that the author did not make to…
So what does it look like if you fudge it by scaling 2 pi down to 6 ?
Bit of a let down when they reveal that plotting integers in general results in the same spiral pattern as primes. So naturally primes as a subset of integers also produce spirals. Seems the title is a bit misleading!
I don't know, for me the fact that this gives you a visualization of a totient is really interesting. Another really interesting idea is to deliberately change the thing which you are rationally approximating; you don't have to rationally approximate π if you don't want to, that's just if you make steps of 1 radian. Make steps of q radians and you get the denominators for rational approximations of q/π. This is used…
https://i.ibb.co/59k5dRT/polarplot.png
Slightly more interesting is what the primes look like when one complete turn is defined as 1, 2, 3... 30 units:
