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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)

#81

Post by Arnt »

kazinator wrote: post #77
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…

> 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.

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

#83

Post by viciousvoxel »

ihatepython wrote: post #60
hinkley wrote: post #53

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.

I'm curious why, even if you were going to approximate it, it would be rounded up to the nearest decimal rather than rounded in the standard way to 3.1?

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

#84

Post by leoff »

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

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

#85

Post by wholinator2 »

BadOakOx wrote: post #65
naniwaduni wrote: post #63

Earlier quoted context omitted.

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).

Well I'm sure it looks trivial to you. But the joys of math often aren't in the difficulty but the discovery. Would your prefer it not have been stated at all, or did you just want to let us know you understood it.

As well, i think the 2k+1 thing is drastically more trivial and not at all equivalent being that all you need to know for 2k+1 is that 2k+1=odd. 4k and especially 6k take a larger generalization and different analytical method and often aren't included in the definition of the primes we learn like 2k+1 (odd) is.

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

#86

Post by Spinnaker_ »

BadOakOx wrote: post #65
naniwaduni wrote: post #63

Earlier quoted context omitted.

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).

Has anyone approached you about writing a math textbook yet? You've already got the tone perfected.

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

#87

Post by jameshart »

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 simple fact blows up into a bunch of even more surprising things that are true of all primes when you follow the algebra to its logical conclusions. Like:

Since this means every pair of twin primes > 3 must be separated by a multiple of 6, so they can be written as 6k+1, 6k-1. That means the product of any pair of twin primes will be of the form 36k^2-1.

In other words take any pair of twin primes, multiply them together, add 1, divide by 36, you are guaranteed to get a perfect square. E.g. 11*13=143, +1=144, /36=4 =2^2.

Or (and this one’s actually a little more complicated because it gets kind of casewise) you can show that the square of any prime (>3) is either one more or one less than a multiple of 24 (which is the product of the 6 and the 4 from the 6k and 4k rules)

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

#88

Post by peter_retief »

misja111 wrote: post #59
peter_retief wrote: post #51

This is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?

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

Why though?

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

#89

Post by jameshart »

onethought wrote: post #25
gnarbarian wrote: post #10

I suspect most functions would make a spiral on a polar graph if you zoomed out far enough.

Well, except as the video points out - primes don't make spiral when you zoom out far enough, they make a "rays" pattern... which then if you zoom out even further becomes a much gentler spiral.

And the fact that pi is irrational means you will continue zooming out getting ever closer to rays, but never actually find rays - you’ll always have a slight spiral.

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

#90

Post by peter_retief »

mik1998 wrote: post #54
peter_retief wrote: post #51

This is one of my favourite mysteries. https://en.wikipedia.org/wiki/Goldbach%27s_conjecture How is it that all even number are the sum of 2 primes and odd number the sum of 3 primes?

We don't know whether it's true.

It is true but unproven.
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