Post · bonfire.cafe

@hipsterelectron@circumstances.run 7 hr. ago

the euclid infinite prime proof is fascinating for many reasons:

the action of first collecting all the known primes q_1, ..., q_k together by commutative multiplication and then adding 1 to it to get the new prime p is a mirror image of the prime field F_p in which p - 1 decomposes into subgroup prime ideals over finite fields with orders corresponding to the prime divisors of p - 1

so at some level even though euclid presumably was not thinking in terms of bits and bytes i do feel that ancient greek geometers were probably thinking closer to the terms of a finite field (consider bisecting an angle!) than anything gauss had ever assumed

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@hipsterelectron@circumstances.run 3 hr. ago

this is also just very specifically the one "NASA" outpost they have that is specifically owned by the CIA. there's like a public autobiography about it somewhere

Floyd, Robert W.; Ullman, Jeffrey D. (1980). "The Compilation of Regular Expressions into Integrated Circuits". NASA Sti/Recon Technical Report N. 81. Fairfax County, Virginia: Ft. Belvoir: Defense Technical Information Center: 12334. Bibcode:1980STIN...8112334F.

@hipsterelectron@circumstances.run 3 hr. ago

also i did spend a good deal of time trying to find an actual living example of euclid's elements and all i could find was some fucking cambridge asshole saying outright that he had written every part of it himself. and of course it all acted like "formalism" was such a huge fucking important thing back in ancient fucking greco-mathematics

@hipsterelectron@circumstances.run 3 hr. ago

ok so bisecting an angle with a compass and straightedge is a pretty direct corollary of the even numbers being a ring closed under multiplication and addition? yes? ok great. now what else

@hipsterelectron@circumstances.run 3 hr. ago

ok so angle bisection corresponds to the ring operation. there's also a very clear higher-dimensional analogy for pythagorean triples and it's just the same thing as the "euclidean" norm?????

@hipsterelectron@circumstances.run 2 hr. ago

i'm reading about the monster group now https://sci-hub.st/10.1038/scientificamerican0680-20 afaict it's another little fairytale the princeton mathematicians are telling themselves

@hipsterelectron@circumstances.run 2 hr. ago

"oooooooo once we factor the monster group we specifically made up in our heads to sound scary we're gonna crack all your isomorphisms or whatever now ooooo"

@hipsterelectron@circumstances.run 2 hr. ago

i can tell you right now i really will need help to avoid trying to prove P < NP based upon modular exponentiation and euclid's least common multiple methodology. it's so incredibly fucking obvious that's the truth now

@hipsterelectron@circumstances.run 2 hr. ago

to be clear, i absolutely have not figured it out yet. to me it's quite clear there are two flaws that have stalled progress:

gauss just asserted uniqueness of prime factorization despite multiplication of integers being an inherently commutative operation. he then decided 1 doesn't count. that's the single thing we need!!! yes 1 divides everything arbitrarily often!!!! euclid didn't have a problem with that!!!
i don't think goldbach's conjecture is true (or at least i don't care). the even integers already have this beautiful ring structure!!!! stop fucking up my beautiful numbers doing their job perfectly!!!!
the primes without 1 and 2 are their own little thing! they all start from one spot then have these cute little directed edges into composites. but each multiplication operation forms a hypergraph edge.

and i think to model it correctly we need a distinct hypergraph edge for every distinct combination of 3+ prime factors to their products. so an edge for 32 => 9, but a separate edge for 33 => 27. we can calculate transitivity later, i just want to model this!

then add infinite edges from 22 => 4, 23 => 8, etc, and keep those edges separate. then add a copy of every 3+ compositional edge for each of the infinite powers of 2 added as a new factor.
then add a loop edge 1 => 1, then 1 x n => n forall n

@hipsterelectron@circumstances.run 1 hr. ago

so we have our hypergraph from hell almost ready to go, but we should make two more operations:

add 1, then take to a power of 2
take to a power of 2, then add 1

but i think that sucks actually though. the point is we need to help get all the 1 divisors who've been trapped in their own ideal for centuries to help us with our little CSPRNG system

@hipsterelectron@circumstances.run 1 hr. ago

but all mathematics aside i feel i can implement a secure and super flexible modular exponentiation primitive now

@hipsterelectron@circumstances.run 1 hr. ago

i need to figure out how to explain to everyone that hashing a directory tree is actually among the deepest questions of modern mathematics

@hipsterelectron@circumstances.run 1 hr. ago

so it's really interesting that within any F_q that 1 is only a fixed point for itself. and that can be readily proven with reference to euclid's cooler and better least common multiple axiom

@hipsterelectron@circumstances.run 1 hr. ago

so wait a second. is it always true that multiplying all primes from 1 to k will necessarily be larger than the k+1'th prime?

@hipsterelectron@circumstances.run 1 hr. ago

well this would certainly have implications for the twin prime conjecture if so. is there any limit we can place on the distance to the next prime at all? can we limit the primes we pick from?

@hipsterelectron@circumstances.run 1 hr. ago

i'm feeling like both goldbach and twin primes might be wrong. just to be a hater

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