Ok going back to the mathematica version that does it the slower primality testing per terms in cell way I am seeing the deviations meaning the gcd triangle is not an exact match to what we have here.

Here is the code on GitHub - it renders it correctly and addresses the issues. Https://GitHub.com/shaunxcode/a-pattern-in-the-primes

From OP: > Interestingly, this same pattern holds for more numbers, and for different intervals.

He's not claiming it holds for _all_ n, just for _many_ n.

I think you may misunderstand what terms are in a cell. For instance in your first example what numbers live there? In that observable you can run cellToTerms(row,n,binomialCoEfficient2,col) to get the terms. You can then chain .map(isPrime) (for reasonable sizes it is slurping in data to make testing faster). Those fns are there for interactivity I have not gotten around to adding yet.

I used range(row * (row - 1) // 2 * n + col, row * (row + 1) // 2 * n + col, row) in Python. For n = 14, row = 13, col = 3, this is

[1095, 1108, 1121, 1134, 1147, 1160, 1173, 1186, 1199, 1212, 1225, 1238, 1251, 1264]

none of which are prime. cellToTerms(13, 14, binomialCoEfficient2, 3) in the observable gives the same list (though no such calls are made when generating the picture).

Those are some pretty basic checks to miss. Perhaps I've misread OP's claim.

I think OP is a bit confused, the post is very ambiguous and on telegram he added more info but still hasn't fully formalized what he's trying to say

OP here - really it’s just a matter of I found a cool thing and wanted others ideas/input. I never had a pretense of a proof or a formalization just a cool pattern that emerges from folding the Ulam Spiral up a certain way. I think we were over zealous in the gcd triangle being an exact correspondence now as it clearly is not for some cells.

The pattern does not exactly match GCD as it turns out - but that makes it more curious actually. Here is the Mathematica code that does work for other values.

Https://GitHub.com/shaunxcode/a-pattern-in-the-primes