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 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
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
and furthermore "the proof that sqrt(2) is irrational", also attributed to euclid, feels really apocryphal. it doesn't extend to anything other than itself. people really play up the idea that fermat invented downwards induction https://en.wikipedia.org/wiki/Proof_by_infinite_descent
Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.
so like this seems completely fabricated right? one of the citations goes https://nrich.maths.org/articles/dangerous-ratio
The murder of Hippasus is a matter of legend, but the secret was real, and certainly dangerous enough to the beliefs of those who knew about it.
so like what if the square root of two was not actually a thing the ancient greek geometers cared about?? and there's another EXTREMELY funny twist: https://en.wikipedia.org/wiki/Well-ordering_principle
so the "well-ordering principle" is what the apocryphal fermat "proof by infinite descent" relies upon. if you've taken math courses you've hopefully been lucky enough to have a mathematician infodump about just how easy it is to fuck up an inductive proof.
and the basic thing i have internalized from the lovely mathematicians bruce hughes and jeremy spinrad at vanderbilt university was that you really absolutely do have to do induction upwards--i.e. you have to construct a more complex instance of your problem directly from a simpler one
consider these two statements about the "well-ordering principle":
Depending on the framework in which the natural numbers are introduced, this (second-order) property of the set of natural numbers is either an axiom or a provable theorem.
i.e. this is like the continuum hypothesis--it inserts an axiom into your logic. if you can't prove something without that axiom, then you are working in a distinct logical system from e.g. ZFC
Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers
ok so this paragraph directly seems to link the existence of a least integer (e.g. +1) to the claim that the real numbers are dense in every interval, which is exactly what i wanted to hear
