It's because we live inside a spiral galaxy that's transitioning into a new galaxy type so we see through a spiral
Why do prime numbers make these spirals? (2019)
61–70 of 141 posts
Re: Why do prime numbers make these spirals? (2019)
#62One 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
Re: Why do prime numbers make these spirals? (2019)
#63One 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…
Isn't 4k+1/4k-1 rule trivial though? If we take any number K=N*4 divisible by 4 and >2, that'd be an even number by definition. The two closest odd numbers on either side would be (K-3), (K-1), (K+1), (K+3). As it happens (K+3) is the same as K(-1) for the next N, and (K-3) is the same as (K+1) for the previous N. So _all_ odd numbers follow this rule. What "4k+1 or 4k-1" says in a roundabout way is that all prime nu…
So is the 6k±1 rule: 6k and 6k±2 are all even, 6k±3 is divisible by 3. You can extend this further: all primes greater than 5 must take one of the forms 30k±1, 30k±7, 30k±11, 30k±13. This is much less exciting, but ... suggestive. (No, not that suggestion, that one isn't actually true.)
For a certain point of view, most of math is trivial corollaries.
(Proof: check.)
Re: Why do prime numbers make these spirals? (2019)
#64One 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…
I have discovered the same thing while practicing prime sieve algorithms back in the day. Such properties of primes are quite useful for optimizing both speed and memory of sieve algorithms. More generally, if you take the first n primes p_1, ..., p_n and define P=p_1*...*p_n , then all primes bigger than P be in the form of P*k +- a , such that a and GCD(P, a) = 1 . In case of n=2 ( P=2*3=6 ), there is this nice pro…
Indeed it's not, it's just anything coprime to P. This explains why there are always ±1s, and in particular why e.g. 210±1/210±209 are viable, even though 209=11*19.
Re: Why do prime numbers make these spirals? (2019)
#65Earlier quoted context omitted.
Isn't 4k+1/4k-1 rule trivial though? If we take any number K=N*4 divisible by 4 and >2, that'd be an even number by definition. The two closest odd numbers on either side would be (K-3), (K-1), (K+1), (K+3). As it happens (K+3) is the same as K(-1) for the next N, and (K-3) is the same as (K+1) for the previous N. So _all_ odd numbers follow this rule. What "4k+1 or 4k-1" says in a roundabout way is that all prime nu…
So is the 6k±1 rule: 6k and 6k±2 are all even, 6k±3 is divisible by 3. You can extend this further: all primes greater than 5 must take one of the forms 30k±1, 30k±7, 30k±11, 30k±13. This is much less exciting, but ... suggestive. (No, not that suggestion, that one isn't actually true.) For a certain point of view, most of math is trivial corollaries. (Proof: check.)
I think this still seems trivial with a 5th grade algebra knowledge.
Additionally, for the 4k+1 / 4k-1 topic, it is just a complicated way of saying 2k+1 (as parent suggested).
Re: Why do prime numbers make these spirals? (2019)
#66One 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…
What's k?
Re: Why do prime numbers make these spirals? (2019)
#67What's the commentary on prime numbers currently? Is it expected that there is some unfound pattern to them? Or will they always remain elusive?
The answer to these questions is the Riemann hypothesis. John Baez just wrote a related paper about "motives" for beginners (undergrads?) [1]. It's worth a read even if you squint at the equations like i do, there's plenty of interesting commentary as well.
Re: Why do prime numbers make these spirals? (2019)
#68One 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…
What's k?
An integer.
Re: Why do prime numbers make these spirals? (2019)
#69One 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…
Trivial correction:
> 6k+3, or 6k-3.
These are the same collection of numbers, you meant to put 6k instead of one of them :-)
Re: Why do prime numbers make these spirals? (2019)
#70One 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…
What's k?
A weird way of writing n?
More seriously, it’s an integer variable. By convention, letters from the middle of the alphabet are used for them (generally n then k).
Here, the commenter uses k because that’s what’s used in the article and that’s what’s used because n is already used to designate the class in the definition of a residue class.