Hey!
Stumbled upon a fascinating little research project: https://p-plus-minus-n-research.netlify.app
The author is exploring how primorials might affect the probability of finding prime numbers in expressions like:
p ± N
where p is a prime, and N is a primorial (i.e., the product of all primes up to a certain limit: 2# = 2, 3# = 6, 5# = 30, 7# = 210, etc.).
The Gist of the Hypothesis:
The idea is that there might be an "optimal" primorial (N₀) for which the probability that either (p + N₀) or (p - N₀) is prime, is maximized. So, the primorial acts as a kind of "filter" for composite numbers, weeding out candidates divisible by small primes.
What the Experiment Shows:
The author ran tests for p up to about 30,000,000 and different primorials N:
If N = q#, then for all small primes r ≤ q, the following holds:
p ± N ≡ p (mod r)
This means that shifting by the primorial doesn't change the remainders modulo small primes. Consequently, it filters out a bunch of composite candidates. It's like a "modular filter" that gives you a better shot at hitting primes.
The Tool:
The site has an interactive JS tool where you can set a range, choose a primorial, and see how many of the p ± N turn out to be prime. It runs right in your browser, so you can play around with it... BUT! I wouldn't put too much trust in it, since JS and big numbers aren't exactly besties
My Two Cents:
The idea is intriguing, and it's great that the author didn't just stick to theory but actually did the computational legwork and provided an open tool. Of course, this isn't a proven law yet, more of an observation within a limited range. It's quite possible the effect weakens for larger p, but still — the direction seems worth looking into.
This could be a nice topic for testing on larger datasets or for searching for a rigorous explanation (if one even exists).
What do you all think? Has anyone come across similar constructions or has experience with such "prime filters"?
I'd be happy to hear your thoughts, verification attempts, or counterexamples!
Stumbled upon a fascinating little research project: https://p-plus-minus-n-research.netlify.app
The author is exploring how primorials might affect the probability of finding prime numbers in expressions like:
p ± N
where p is a prime, and N is a primorial (i.e., the product of all primes up to a certain limit: 2# = 2, 3# = 6, 5# = 30, 7# = 210, etc.).
The Gist of the Hypothesis:
The idea is that there might be an "optimal" primorial (N₀) for which the probability that either (p + N₀) or (p - N₀) is prime, is maximized. So, the primorial acts as a kind of "filter" for composite numbers, weeding out candidates divisible by small primes.
What the Experiment Shows:
The author ran tests for p up to about 30,000,000 and different primorials N:
- The maximum density of prime candidates was achieved with N = 29# — roughly 6.28 times higher than the expected standard prime density.
- However, the maximum efficiency (i.e., the actual proportion of p for which p ± N turns out to be prime) was observed with N = 19#.
- Beyond that, increasing N boosts the density, but the efficiency starts to drop — which looks like some kind of trade-off.
If N = q#, then for all small primes r ≤ q, the following holds:
p ± N ≡ p (mod r)
This means that shifting by the primorial doesn't change the remainders modulo small primes. Consequently, it filters out a bunch of composite candidates. It's like a "modular filter" that gives you a better shot at hitting primes.
The Tool:
The site has an interactive JS tool where you can set a range, choose a primorial, and see how many of the p ± N turn out to be prime. It runs right in your browser, so you can play around with it... BUT! I wouldn't put too much trust in it, since JS and big numbers aren't exactly besties
My Two Cents:
The idea is intriguing, and it's great that the author didn't just stick to theory but actually did the computational legwork and provided an open tool. Of course, this isn't a proven law yet, more of an observation within a limited range. It's quite possible the effect weakens for larger p, but still — the direction seems worth looking into.
This could be a nice topic for testing on larger datasets or for searching for a rigorous explanation (if one even exists).
What do you all think? Has anyone come across similar constructions or has experience with such "prime filters"?
I'd be happy to hear your thoughts, verification attempts, or counterexamples!
