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An Interesting Pattern in the Prime Numbers: Parallax Compression

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Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#71

Post by mathythrowaway »

anderskaseorg wrote: post #68
mathythrowaway wrote: post #65

Earlier quoted context omitted.

Their picture is correct, but only because they are doing something more ridiculous than this. Every one of the squares in their picture represents 75 numbers, not 5 numbers.

Click on the definition of T, and you see exactly the A054521 formula. T = (n, k) => { return (GCD(n, k) == 1) ? 1 : 0; } The given code doesn’t use primes, primesSet, or isPrime at all.

But that does produce a picture identical to theirs. Because they are using 75 numbers per square, they do end up hitting at least one prime in each box with (n,k)=1 for the first 75 rows. This pattern won't continue much longer, which is why they've conveniently stopped the picture at that point.

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#72

Post by szemet »

Whitespace wrote: post #64

It wasn't exactly clear to me when I read the article, so here's my explanation: - take a line of integers, color them black if prime, red otherwise - hexagonally arrange them in a spiral (similar to Ulam's spiral) - cut the hexagon into six equilateral triangles - overlap the triangles (rotate where necessary) - if any pixels are black, color the whole thing black, if none are black, color it red - interesting patte…

Just check the numbers listed in one of the referenced articles, it is easy to expand it by induction to numbers greater than 6: https://beta.observablehq.com/@montyxcantsin/unwinding-the-u...

It also helps to understand my other comment about the explanation of the pattern and GCD equivalence...

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#73

Post by _rpd »

mathythrowaway wrote: post #71
anderskaseorg wrote: post #68

Earlier quoted context omitted.

Click on the definition of T, and you see exactly the A054521 formula. T = (n, k) => { return (GCD(n, k) == 1) ? 1 : 0; } The given code doesn’t use primes, primesSet, or isPrime at all.

But that does produce a picture identical to theirs. Because they are using 75 numbers per square, they do end up hitting at least one prime in each box with (n,k)=1 for the first 75 rows. This pattern won't continue much longer, which is why they've conveniently stopped the picture at that point.

The claim is that using N numbers per square will result in the pattern holding for N rows, for arbitrarily large N.

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#74

Post by [deleted] »

mathythrowaway wrote: post #71
anderskaseorg wrote: post #68

Earlier quoted context omitted.

Click on the definition of T, and you see exactly the A054521 formula. T = (n, k) => { return (GCD(n, k) == 1) ? 1 : 0; } The given code doesn’t use primes, primesSet, or isPrime at all.

But that does produce a picture identical to theirs. Because they are using 75 numbers per square, they do end up hitting at least one prime in each box with (n,k)=1 for the first 75 rows. This pattern won't continue much longer, which is why they've conveniently stopped the picture at that point.

[deleted]

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#75

Post by drdeca »

I am a little confused. When it says that each square corresponds to n sequential numbers, does it mean that the first square is the first n, and the second square (which is on the second row) is the first n after those, and the third square (also in the second row) has the first n after those, and so on,

Or, do the regions overlap?

I tried to look at the first 4 rows for n=5, but did not see the pattern depicted (each interval had a prime in it).

Am I interpreting what is meant by the blocks incorrectly, or does the pattern not work for small enough n?

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#76

Post by chickenstrips »

drdeca wrote: post #75

I am a little confused. When it says that each square corresponds to n sequential numbers, does it mean that the first square is the first n, and the second square (which is on the second row) is the first n after those, and the third square (also in the second row) has the first n after those, and so on, Or, do the regions overlap? I tried to look at the first 4 rows for n=5, but did not see the pattern depicted (ea…

For fixed n, a block (x,y) contains the numbers n/2y^2-n/2y + yz + x for all 0 <= z < n.

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#78

Post by drdeca »

brador wrote: post #62
kibwen wrote: post #37

These runes would serve excellently as a written alphabet for an alien species in fiction (or maybe for modrons or other Lawful creatures in D&D).

I'd like to see the triangle fractal as a starting state for Conways "Game of Life".

As displayed in the image, every other row is offset by half a square from being on a grid.

Maybe if you shifted everything to the left to line it up, but

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#79

Post by anderskaseorg »

mathythrowaway wrote: post #71
anderskaseorg wrote: post #68

Earlier quoted context omitted.

Click on the definition of T, and you see exactly the A054521 formula. T = (n, k) => { return (GCD(n, k) == 1) ? 1 : 0; } The given code doesn’t use primes, primesSet, or isPrime at all.

But that does produce a picture identical to theirs. Because they are using 75 numbers per square, they do end up hitting at least one prime in each box with (n,k)=1 for the first 75 rows. This pattern won't continue much longer, which is why they've conveniently stopped the picture at that point.

Can you explain how that picture would be produced with 75 numbers per square? The generalization that seems obvious to me already fails in both cells of row 2, because 75 is odd:

row 1, col 1: [1, 2, 3, …, 75], has primes

row 2, col 1: [76, 78, 80, …, 224], no primes

row 2, col 2: [77, 79, 81, …, 225], has primes

Re: An Interesting Pattern in the Prime Numbers: Parallax Compression

#80

Post by clort »

ssewell wrote: post #43
_rpd wrote: post #33

Earlier quoted context omitted.

The claim is that it is the same pattern independent of the number of integers per dot.

If the numbers of integers-per-dot was set to 1, the sequence would simply highlight prime numbers as they progress through the pyramid shape since “Parallax Compression” would no longer apply.

Yet they say that this only seems valid for the number of rows equal to the number of integers-per-dot that was used. So, it would indeed highlight with reasonable certainty all the primes in the first row. (I am not a mathematician but I think that is "1" ?)
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