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Why do prime numbers make these spirals? (2019)

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Re: Why do prime numbers make these spirals? (2019)

#91

Post by eternalban »

westcort wrote: post #34

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…

That is because 6 is a priomorial. For any primorial p', k*p' +/- 1 will result in a number relatively prime to p', some of which are absolute primes.

The key to understanding primes is in relative primes and reduced residue sets. All patterns in (higher) primes (absolute) are generated by the members of RRS of smaller primes. This includes the clusters, such as twins, triple, quadruple, ..., primes. RRSs also hint [imo] at intimate connection between complex numbers and primes.

https://en.wikipedia.org/wiki/Primorial

https://en.wikipedia.org/wiki/Coprime_integers

https://en.wikipedia.org/wiki/Reduced_residue_system

https://en.wikipedia.org/wiki/Root_of_unity

Re: Why do prime numbers make these spirals? (2019)

#92

Post by Vvector »

leoff wrote: post #84
westcort wrote: post #34

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…

>all primes greater than 3 are of the form 6k+1 or 6k-1 what's the value in using this "formula"? We could also keep extending this rule, and say that all primes greater than 5 are of the form 30k±1, 30k±7, 30k±11, or 30k±13. Or go further by multiplying coefficient of X with the next primes

30k±1 = 6x±1 where x = 5k and so on. Why use four formulas when one will suffice?

Re: Why do prime numbers make these spirals? (2019)

#93

Post by misja111 »

peter_retief wrote: post #88
misja111 wrote: post #59

Earlier quoted context omitted.

The odd number sums follow from the fact about the even numbers.

Why though?

If you can make any even number with 2 primes, you can make an odd number by subtracting one odd prime from the odd number first (e.g. 3) and the resulting even number with 2 more primes.

Re: Why do prime numbers make these spirals? (2019)

#94

Post by hn_throwaway_99 »

CyberDildonics wrote: post #33
hn_throwaway_99 wrote: post #21

Earlier quoted context omitted.

Hence, the video...

Hence the video what? It's polar coordinates making the spiral, not prime numbers.

Hence the video making it extremely clear pretty much right off the bat that yes, any integer plotted that way will form a spiral. The video then goes into a very interesting discussion IMO into why you see the specific patterns of spirals and rays with the primes.

Re: Why do prime numbers make these spirals? (2019)

#95

Post by rvbissell »

Arnt wrote: post #81
kazinator wrote: post #77

Earlier quoted context omitted.

> the discovery that ... 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 c…

My textbook, at least, spent its space on the important axioms, theorems and corollaries. There were some easily-rediscovered results there, but mostly its pages described stuff that wasn't trivial to me. And that's why it's one of the five books I have kept in the decades since.

Care to share the name of that textbook?

Re: Why do prime numbers make these spirals? (2019)

#98

Post by Arnt »

rvbissell wrote: post #95
Arnt wrote: post #81

Earlier quoted context omitted.

My textbook, at least, spent its space on the important axioms, theorems and corollaries. There were some easily-rediscovered results there, but mostly its pages described stuff that wasn't trivial to me. And that's why it's one of the five books I have kept in the decades since.

Care to share the name of that textbook?

Authors called Edwards and Penney IIRC, but it's lent out this year.

Re: Why do prime numbers make these spirals? (2019)

#99

Post by oluckyman »

CGamesPlay wrote: post #48
westcort wrote: post #34

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…

This is neat and I never noticed this. For 5, I guess the linear component would be (2*3*5)k, but it isn't as interesting or useful because the constant component would be +/- 1, 7, 11, or 13. This method feels like it's basically a "higher order prime sieve". [append] Oh, and because this pattern (of 1 always being one of the constants) carries out for arbitrarily large linear coefficients, that also explains the "t…

A “higher order prime sieve” indeed: https://en.wikipedia.org/wiki/Sieve_of_Pritchard

Re: Why do prime numbers make these spirals? (2019)

#100

Post by zamadatix »

Vvector wrote: post #92
leoff wrote: post #84

Earlier quoted context omitted.

>all primes greater than 3 are of the form 6k+1 or 6k-1 what's the value in using this "formula"? We could also keep extending this rule, and say that all primes greater than 5 are of the form 30k±1, 30k±7, 30k±11, or 30k±13. Or go further by multiplying coefficient of X with the next primes

30k±1 = 6x±1 where x = 5k and so on. Why use four formulas when one will suffice?

For the same reason you'd use 6x±1 instead of 1+2n or, more basically, n - higher check hit rate at the cost of more complex checking rules.
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