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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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
@hipsterelectron what
7, 17, 19, 31, 37
@hipsterelectron wait wait wait what about
27 = 2*13 + 1
