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The rendering was for n=74 in fact, not for n=75. The pattern does seem to recur for even numbered n values and for those we think it matches GCD (where cells

The rendering was for n=74 in fact, not for n=75. The pattern does seem to recur for even numbered n values and for those we think it matches GCD (where cells contain an even number of integers, and we render for the same even number of rows). But this has not been proved yet. However when n has odd values, the pattern does not always match GCD, it turns out. This is interesting and means that it may be a less “trivial” pattern in fact. See the Mathematica notebook, which correctly renders it for any value. Https://GitHub.com/shaunxcode/a-pattern-in-the-primes

airesearcher 8y ago
See this animation by Ian Rust showing how the even values of n approach the GCD pattern as the values increase: https://streamable.com/l7r96 https://streamable.com/l7r96 However, for odd values of n, it does not match GCD. So it's not simply a drawing of the GCD pattern.
- macawfish 8y ago
  This is fascinating!
- svat 8y ago
  Well, for obvious reasons: when n is odd, and the row R is even (specifically, 2 mod 4, i.e. of the form 4k+2 for some k) and the column C is odd, then even if gcd(R, C) = 1, the set being considered contains only even numbers — note that it starts with R(R-1)N/2+C which will be even as R(R-1)/2 is odd and so is C - so you won't have any primes in it. (That is, you'll have a red cell, even where the gcd pattern has a black one.) And when n is even, there's also a heuristic argument that there are only finitely many counterexamples; i.e. you should expect to find the pattern from your process exactly match that from just the gcd, beyond a few counterexamples: https://news.ycombinator.com/item?id=17106648 https://news.ycombinator.com/item?id=17106648
- leereeves 8y ago
  > the even values of n approach the GCD pattern as the values increase Meaning that the chance of finding a prime in these sequences of N numbers approaches one, after eliminating the cases that make primes impossible (n odd and r=2 mod 4, gcd(r,c) > 1). Is there any deeper meaning to that, or is it simply that the probability of finding a prime in any sequence of n numbers approaches one as n increases, excluding sequences in which primes are impossible?

airesearcher 8y ago

See this animation by Ian Rust showing how the even values of n approach the GCD pattern as the values increase: https://streamable.com/l7r96 https://streamable.com/l7r96 However, for odd values of n, it does not match GCD. So it's not simply a drawing of the GCD pattern.

macawfish 8y ago
This is fascinating!
svat 8y ago
Well, for obvious reasons: when n is odd, and the row R is even (specifically, 2 mod 4, i.e. of the form 4k+2 for some k) and the column C is odd, then even if gcd(R, C) = 1, the set being considered contains only even numbers — note that it starts with R(R-1)N/2+C which will be even as R(R-1)/2 is odd and so is C - so you won't have any primes in it. (That is, you'll have a red cell, even where the gcd pattern has a black one.) And when n is even, there's also a heuristic argument that there are only finitely many counterexamples; i.e. you should expect to find the pattern from your process exactly match that from just the gcd, beyond a few counterexamples: https://news.ycombinator.com/item?id=17106648 https://news.ycombinator.com/item?id=17106648
leereeves 8y ago
> the even values of n approach the GCD pattern as the values increase Meaning that the chance of finding a prime in these sequences of N numbers approaches one, after eliminating the cases that make primes impossible (n odd and r=2 mod 4, gcd(r,c) > 1). Is there any deeper meaning to that, or is it simply that the probability of finding a prime in any sequence of n numbers approaches one as n increases, excluding sequences in which primes are impossible?