Hello! I wrote this simple prime number grid visualiser last night, just for fun. It is inspired by the "Show HN" post https://news.ycombinator.com/item?id=44888548 that I stumbled upon a few days ago.
My tool uses the Miller-Rabin primality test with prime bases drawn from https://oeis.org/A014233 to determine whether a number is prime. This allows it to handle numbers up to 3317044064679887385961980.
I think another interesting feature would be if you could change the number base to 16 or some other base, I'm really curious if the pattern would change.
lblume · 5s ago
Numbers are prime irrespective of base.
Tepix · 58m ago
Great visualization! Can you please add a on-mouse-over so when i hover my mouse over a dot i can see the prime number it represents?
Would we see new patterns emerge if the number of columns increases per row by X (X being constant or perhaps prime numbers ;-) )?
susam · 11m ago
Adding mouseover text to every prime number slows down rendering on large grids (say, with a million or more numbers). So mouseover text is available as an optional feature. You can toggle it using the 't' button at the top: click once to enable the text, and click again to disable it.
camillomiller · 1h ago
It's so cool actually!
You actually sent me on a rabbit hole trying to visually look for patterns :D
But I guess the discretionality with which you can organize in rows and columns makes mine quite a pointless excercise :D
jona-f · 1h ago
If you select 30, 60 or 90 columns you get the clearest patterns. It kinda seems that the more divisors the number of columns has, the clearer the vertical clusters are. And somehow 30, 60 and 90 stand out. Number theory is so weird. I expected more randomness.
susam · 1h ago
The reason vertical clusters appear in these examples is that all your chosen numbers are multiples of 6. A prime number greater than 3 leaves a remainder of either 1 or 5 when divided by 6. In other words:
For all primes p greater than 3, p ≡ ±1 (mod 6).
Therefore, when the total number of columns is a multiple of 6, all primes except 2 fall into the same columns, namely 1, 5, 7, 11, 13, 17 and so on.
jacobtomlinson · 1h ago
I just set the column width to 6 to verify this for myself. What a neat tool!
Daub · 14m ago
Very cool. Go from 30 to 31 to see this ‘pattern’ twist in on itself.
ainiriand · 25m ago
Go for 258 and be ready to get your mind blown.
vintermann · 32m ago
Fun to see that prime numbers of columns causes stripy patterns, and some stripe to the left and some stripe to the right. Probably some deep number theoretic reason for that.
kitd · 36m ago
Really interesting, stepping up and down the "cols" number, seeing the dots align at certain key points, especially at multiples of 30.
krige · 33m ago
There's also a cool effect where incrementing columns by one has the dots align diagonally, and then vertically multiple times.
anthk · 6m ago
I'd love this but in SDL+GL allowing to scale up and down an image.
Or better, a command to write an XBM/XPM image and then I'd convert it to any format I like.
rmrfchik · 2h ago
Nice patterns are reveals when cols is prime.
ethan_smith · 1h ago
This happens because when columns = p (prime), numbers in each column share the same remainder mod p, creating visible diagonal patterns as multiples of p are eliminated from primality.
scotty79 · 34m ago
Almost all of these patterns that you see don't really come from primes. If you display numbers not divisible by first 100 natural numbers you get pretty much the same picture.
fendy3002 · 19m ago
it's interesting that for 6 cols only the 1st and 5th column has value, ignoring first row.
martinclayton · 1h ago
Hours of fun (stimulation?) to be had...
Try these shapes: 100x113, then 100x114, then 100x115, the "patterns" swing from slant down, to vertical, to slant up.
I'd love this (even more) with some animation and colo(u)r options.
susam · 48m ago
This was just a quick experiment I hacked together last night in my free time, so the tool is still quite bare bones. If you're on a desktop browser and don't mind opening the developer console, you can run this little snippet to animate the grid:
In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
> In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
When written down as a string of digits, log n is another way to say 'proportional to the number of digits'.
The number of digits grows fairly slowly, thus also the 'probability' of a number being prime drops very slowly.
That’s what ‘everybody’ thinks. I think that’s from reading so much about them being hard to find. They aren’t hard to find, though, it’s (as far as we know) hard to recognize integers as being primes.
There are more prime numbers than there are squares of integers.
throw310822 · 47m ago
> I think that’s from reading so much about them being hard to find
More from the fact that each prime number makes all its multiples non-prime, so you'd expect this would accumulate quickly in making primes an increasingly rare find. Which is the case, but slower than intuition suggests.
eru · 1h ago
> They aren’t hard to find, though, it’s (as far as we know) hard to recognize integers as being primes.
Depends on what you mean by 'hard'. It's easy in the sense that we have algorithms to decide whether a number is prime or composite that take time polynomial in the space it takes to write down your number (ie polynomial in log n).
mickeyp · 1h ago
I really love susam's blog posts and curiosity. I highly recommend that people check out his site for more of his insights.
nyc111 · 16m ago
I tried but his pages do not have links to a home page or other posts
dev0p · 33m ago
Editable size pls? I wonder if this could be visualized in 3 dimensions...
vim-guru · 1h ago
This works surprisingly well for logo design. Cool concept
quijoteuniv · 1h ago
Wow! I see the pattern now! ;) … nice work
agnishom · 1h ago
Since this is in a grid, how about visualizing Gaussian primes instead?
My tool uses the Miller-Rabin primality test with prime bases drawn from https://oeis.org/A014233 to determine whether a number is prime. This allows it to handle numbers up to 3317044064679887385961980.
For example, https://susam.net/primegrid.html#3317044064679887385961781-2... shows the upper limit of the numbers this tool can check. The three circles displayed there represent the following prime numbers:
I hope this is fun for you too!Would we see new patterns emerge if the number of columns increases per row by X (X being constant or perhaps prime numbers ;-) )?
You actually sent me on a rabbit hole trying to visually look for patterns :D But I guess the discretionality with which you can organize in rows and columns makes mine quite a pointless excercise :D
For all primes p greater than 3, p ≡ ±1 (mod 6).
Therefore, when the total number of columns is a multiple of 6, all primes except 2 fall into the same columns, namely 1, 5, 7, 11, 13, 17 and so on.
Try these shapes: 100x113, then 100x114, then 100x115, the "patterns" swing from slant down, to vertical, to slant up.
I'd love this (even more) with some animation and colo(u)r options.
In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
I have a small section about this at https://susam.net/journey-to-prime-number-theorem.html#prime... if you want to read more about this.
See also: https://en.wikipedia.org/wiki/Prime_number_theorem
> In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.
When written down as a string of digits, log n is another way to say 'proportional to the number of digits'.
The number of digits grows fairly slowly, thus also the 'probability' of a number being prime drops very slowly.
https://en.m.wikipedia.org/wiki/Prime_number_theorem
There are more prime numbers than there are squares of integers.
More from the fact that each prime number makes all its multiples non-prime, so you'd expect this would accumulate quickly in making primes an increasingly rare find. Which is the case, but slower than intuition suggests.
Depends on what you mean by 'hard'. It's easy in the sense that we have algorithms to decide whether a number is prime or composite that take time polynomial in the space it takes to write down your number (ie polynomial in log n).