one epiphany i had a few hours ago is that the reason q = 2p + 1 always generates another prime is emphatically not so much the 2p term, but the +1. the integer +1 is essentially the point at infinity for the integers
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@hipsterelectron what
7, 17, 19, 31, 37
@hipsterelectron wait wait wait what about
27 = 2*13 + 1
every prime number > 1 describes a period (which can easily be analogized to a synchronization of orbits) of some process. composites > 1 match up against multiple shorter periods. when you transform these to operations in a finite field F_q by performing integer arithmetic (mod q), you can see that addition terms like a * q + k correspond to phase offsets in a periodic orbit
the euclidean proof of the infinitude of primes (the one that i'm aware of) merely states:
[by contradiction]
- say the cardinality of the set of natural primes > 1 |{P_N}| is finite
- let
q =( \Pi_{p_i \in P_N} p_i) + 1.\Pi_{p_i}must converge to a finite product- yet
q"clearly" cannot be divided by any member ofP_N- QED
this still kinda dissatisfied me (how do we know +1 is special like that?) until i considered the periodic orbits analogy
@hipsterelectron 1 is special because it has no prime factors and is smaller than every prime, hence step 4
see this is also why the "natural" numbers are actually incredibly un natural imho. whereas finite fields and the complex plane naturally encode periodicity and phase offsets, the set [1,∞) is completely "unrolled" by comparison. what does it mean to add a number at all?
i had a lot of thoughts in response to this correct post https://types.pl/@amy/117203193831178704
as of late i have begun to feel that the imaginary component of so-called "complex" numbers is strikingly similar to irrationality and indeed that the "unique" factorization into prime integers or polynomials with integer coefficients is much closer to a parlor trick than a rigid basis for classification
like the irrationals are only secondarily "irrational" and more simply "antipolynomial" i feel like? i read an attempted motivation for the irrationality of
sqrt(2)arising from euclidean-era geometry recently but i didn't really buy it since i don't feel "exactness" or "equality" is really a "natural" concept outside of the cartesian plane, which itself feels remarkably circular in definition?like if even the prime numbers refuse to decompose their own sequential structure into any kind of rhythmic progression, maybe that's because "solutions among the rationals" is requiring the answer be sufficiently *un*interesting?
learning much more about group theory recently and i totally do accept that the generating graph of modular exponentiations in a finite field is intrinsically meaningful to study even if it's not a "nice" answer for my purposes. but finding that finite fields
F_qfor primes p < q s.t.q = 2p + 1are not just a meme way to generate new primes at will (which i didn't know you could do????) but are also optimally resistant to computational factorization attacks on the discrete log problem has me feeling like twin primes just aren't very interesting?like if generating the next prime is unsolvable but just finding a later prime is both simple and deeply meaningful, maybe that's because a difference of integer
+kis not intrinsically meaningful except as a phase offset wiithin a periodic context?like forget
sqrt(-1), the really unnatural "point at infinity" seems to be the integer+1, since that's precisely what's necessary for the infinitude of primes!!!
if we add +2 to a number n \in [1,∞), we're adding a phase offset that (if repeated) would eventually generate periodic behavior, and similarly for all n >= 2. but what the hell does adding +1 mean?
similarly, repeated multiplication of a by 1 (say, in a finite field F_q will never generate any value distinct from a, even "infinitely" many times
so the fact that we can just "add 1" to the result of multiplying primes in N = [1,∞) is beginning to feel circular
and by the way, the cartesian plane makes absolutely no goddamn fucking sense. it is the most artificial construct. a cross-hatched lattice with perfect right angles at all mesh intersections? i don't believe you can even define the cartesian plane without incorporating a hidden circular assumption involving the limits of sequences tending towards infinity
and so are "polynomials". think about this for a moment (because it shook me personally)--the square root of 2 is a number that clearly exists, because it's what you get from a right triangle with equal sides. yet we call it "irrational"?
and the reason we even care about sqrt(2) is because it's what arises when we attempt to "solve" a polynomial representation of the simple right triangle
@hipsterelectron "irrational" as in "not a ratio", not "illogical"
@eevee but the real reason irrationals are "interesting" or at all worthy of naming as a class (taking the square root of 2 as an example) is because they describe a class of solutions to finite-degree integer polynomials with rational coefficients. the fact that sqrt(2) exists as a thing to name is because of how it falls out of the solutions to the quadratic equation
@hipsterelectron that is also not true. the vast majority of irrationals are not solutions to any polynomial. they are, by and large, /not/ interesting, which is why they're named "not those other numbers"
what you're talking about are algebraic numbers
@eevee those are fighting words!!! i think the integers are the boring ones!!! unlike a finite field which loops over itself, the integers have been unfurled across the number line and taped down against the rigid axis of the cartesian plane
@eevee er, sorry, i meant the polynomials are the really highly rigid and restricted class. integers are fine we get along
@eevee the reason complex numbers make sense at all is because the number 1 actually forms the origin point where two bases meet, and that's where the function f(x) = 1/x (not a polynomial) comes in
@eevee and then we just suddenly get this number e which doesn't really mean anything except as the perpendicular tangent line to the basis vectors of these two infinite spaces from (0,1] through 1/x and [1,∞) through f(x) = x, where the continued fraction representation (while being necessarily transcendental) precisely mirrors the unsolvable infinitude of the primes themselves!
https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
the infinite sum of 1/n^s is equal to the infinite product of 1/(1-p^{-s}) for p prime??????
that's what fucked me up earlier. the structure of the primes is not (imho) their individual jitters in the sums and differences, but how they eventually form the generating set of all the spaces in between their periodic orbits when taken out to infinity
@eevee it always really pissed me off the way a certain type of quantum physics fuckboy (not you, you're really cool) finds the regular structure of differentiable wave equations too regular and too easily solvable, and consequently missing the forest for the trees in describing absurdities like a qubit because the concept of fixed integer relationships being the result of a standing wave of precise periodicity would mean they'd have to map between the smoothly differentiable wave equations and their own little sublanguage of mathematics
@eevee this is also very important to me personally because if you invert that worldview you find you can in fact model arbitrarily massive or unfathomably minute objects without needing a supercomputer and several trillion dollars to burn in order to do real scientific research. and i of course cannot prove it yet but it seems that in fact the difficulty of stating whether P = NP is more of a statement about polynomials being a very fiddly set of objects to make statements with than anything else
@eevee and if P < NP (as i now believe it must), then we provably have safe cryptography from the discrete log problem holding fast against any fascist state investing trillions in supercomputing. the double ratchet cryptosystem computes two new discrete logs every single message! and that makes it resilient to "adversarial randomness" https://eprint.iacr.org/2020/148, a concept which had to be coined to explain how this relatively simple mathematical object designed by a queer transmasc who majored in classics in undergrad was more effective than the entire history of academic cryptography!
and notably, much like how the additive generating function of the integers has a perpendicular component in the continued fraction of 1/x, the double ratchet cryptosystem incorporates both a symmetric and asymmetric ratcheting step, which act so very much like an addition and multiplication operation composed together!
@eevee so i mean i guess you would be exactly right then about the algebraic numbers having a lot of meaning. i think my major issue is the conception that that arises from factoring them downwards into simplest form. algebraic roots of polynomials of larger degree have no closed form solution and that's a good thing!!!
if the diffie-hellman approach with the discrete log problem does result in a proof of P < NP, i think it would come from seeing that the problem as stated is essentially underconstrained unless you know the precise input x your target had in mind when they computed a^x (mod p).
in other words, the mind is not just the last safe place, it is also the first, and that's more than enough
