my dealer: got some straight gas 🔥😛 this strain is called "number theory" 😳 you'll be zonked out of your gourd 💯

me: yeah. whatever. i don’t feel shit.

5 minutes later: dude i swear i just saw some twin primes outside the window

my buddy euclid pacing: En = pn# + 1 where pn# is the nth primorial (the product of the first n prime numbers)

countability

"DMT_18," digital + acrylic, April 20, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

So here's an interesting fact I've learned while researching these things called Wolstenholme Primes. See they're related to the binomial coefficient and thus pascals triangle. We've probably all seen sierpinski's triangle in here (if not circle all the odd numbers). But here's something else interesting. If you look at the triangle, you'll see that for prime numbers, every element is divisible by that prime number. Look at 11 for example.

55, 165, 330, and 462 are all divisible by 11. This has to do with the binomial coefficient definition, in which you'll see that if unless we choose either all the elements, or none of them, the factor of our prime number is guaranteed not to cancel out from the numerator's factorial.

We can then use the fact that the sum of all the elements in the nth row of pascal's triangle adds up to 2^n. Once we remove the 1's on each end of the triangle (hence the minus 2) we know that for prime numbers p that all elements are divisible by p, their sum is as well!

Now, I don't know off the top my head if this is only true for prime numbers. It isn't immediately obvious imo. However, if this is strong enough to be an if and only if, then I think I have independently discovered perhaps the *worst* way to test for primality lmao, scaling exponentially instead of the standard logarithmically.

Can anyone come up with a function that always sends algebraic numbers to algebraic numbers, but has a local max or min at a transcendental number?  I think there should be one but I can’t come up with one.  

i'm tired of only having uppercase numbers

G. H. Hardy – Scientist of the Day

G.H. Hardy, an English mathematician, was born Feb. 7, 1877, with the first names Godfrey Harold, which he never used, preferring "G.H." 

read more...

what kind of research are you doing for your phd thesis? (apologies if you've posted about this before)

So the stuff I’m working on right now isn’t quite a thesis (not yet anyway, it might turn into one later). I can’t get into too many details without doxxing myself, but it involves objects called modular forms, which appear all over the place in mathematics. They’re very analytic objects, coming from complex analysis, having Fourier series, etc. But they also satisfy a bunch of symmetries (dictated by the so-called full modular group). This algebraic structure means that the coefficients of their Fourier series actually encapsulates a lot of arithmetic information.

Now, the combination of the analytic and algebraic niceness of things means that there’s only a handful of them (more specifically they form a finite dimensional complex vector space). This means if you have something that you can associate a modular form to, that actually tells you a lot about what it could possibly be. This association was at the heart of what became the proof of Fermat’s Last Theorem.

would anyone be interested in an upper level mathematics discord? I know there's math and hw servers out there but I was thinking it may be nice to help each other with let's say everything past precal ?

The anecdotal god be like about Quantum mechanics: Let's put a pinch of number theory in physics! Make the logical fuckery extra spicy!

"DMT_9" digital + acrylic, April 11, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

Back to dragon curve shaders

I'm once again playing with dragon curve shaders, this time with a unity implementation. Friendly reminder this is an entirely parallelized gpu implementation, something which is not an easy task to derive!

We've made some serious optimizations this time with our newfound number theory interpretation, though it cant handle as large of numbers as my old matrix multiplication interpretation, because I haven't hard-coded the higher precision types. Still this should be able to handle ~31 iteration dragon curves, and it will do so re-generating it in real time. I've yet to profile it but Just looking at the code it should be much cleaner.

In fact, I can fit the fragment shader and Dragon method into a single screenshot, and look at how clean it looks! Granted we're missing some bells and whistles from the old version, but I'm still happy with this. The only new math I haven't posted about before here is the way I'm shading the curve itself.

Basically, we keep track of some running index. Turns out, the direction you fold the curve alternates, then the direction you fold each of those new little sections alternates, and so on. Basically if you keep track of the turns in binary (0 for turning ccw, 1 for turning cw) you just get a number ranging from 0 to however many segments are on the curve. Divide it by the number of segments and you get a normalized position from 0-1. There's a little bit of artifacting because my method only applies to midpoints and I haven't handled all the other needed calculations, but it works well enough

A new post up on my blog!  Last time we talked about the algebraic numbers, and how just wanting to solve simple equations can create a ton of different numbers.  But they don’t get us everything.

So this time we start off with the idea of measurement, and wind up inventing the real numbers.  The real numbers are weird.  Real weird.  But they show up when we start asking questions about size or measurement.  And in part 3, we’ll see they’re exactly the right way to do calculus.

i am so confident that will disappeared at 9pm in s1. i cant 100% prove it but I just Know. he left the wheeler house “a bit after 8,” according to karen, and we see the number 9 associated with the colour yellow in S4, whereas 6 is associated with blue (and those two numbers, interestingly, are literal upside-down versions of eachother- I’m gonna make a post about it once I go back through the seasons and look for more connections)