For learning the theory behind quantum computing, I usually recommend Watrous's lecture notes [1] - they start out by immediately giving a helpful analogy to ordinary probabilistic computation.
The online tutorial [2] is a good followup, especially if you want to understand Clifford gates / stabilizer states, which are important for quantum error correction.
If you have a more theoretical bent, you may enjoy learning about the ZX-calculus [3] - I found this useful for understanding how measurement-based quantum computing is supposed to work.
There is no general procedure for computing upper bounds on busy beaver numbers (this can be proven). We haven't even come close to enumerating all of the interesting six-state Turing machines, so right now we don't even have a wild guess for an upper bound on BB(6).
I've only skimmed the paper, but this looks very nice: the construction is very simple (aside from the precise choices of the parameters), just the analysis to show that it works is difficult.
(I bet the construction can be refined - it feels like there is a semidefinite programming problem lurking in the background, so there is probably a way to mindlessly optimize things with an SDP solver once the proof technique is rephrased a bit.)
Unfortunately no, ZFC isn't good enough to capture arithmetical truth. The problem is that there are nonstandard models of ZFC where every single model of second-order PA within is itself nonstandard. There are even models of ZFC where a certain specific computer program, known as the "universal algorithm" [1], solves the halting problem for all standard Turing machines.
ZFC allows models of second order PA and proves that those models are all isomorphic. Within each model of ZFC there is no such thing as a nonstandard model of second order PA. One can only think it is nonstandard by looking from outside the model, no? What theorem of second order PA is ZFC unable to prove?
This is similar to how there are countable models of ZFC but those models think of themselves as uncountable. They are countable externally and not internally.
Indeed yes. But in a sense within ZFC one can say what N is given the categorical nature of second order PA. Each model of ZFC will have, up to isomorphism, one model of N.
Speaking as someone from the math community: 90% of the time, when we get a request like this, there is some form of mental illness involved. We aren't psychologists, so we tend to handle that really poorly. Well, let's take a look...
- It looks like the first block of stuff here is some equalities in Q[√5] which you have checked either numerically or symbolically (I haven't checked any of them in detail, but I doubt you made any mistakes here).
- The first claimed breakthrough is a representation of the fundamental unit of a Pell equation as a linear function of √5, but I'm pretty sure that this has been known for at least 200 years (maybe much more than that, I'm not an expert in history of math). I guess it's supposed to be new because you are using these other constants T, J, K instead of √5, but mathematically this is just a roundabout way of writing down the same thing, since they are all linear functions of each other.
- We then have the standard formulas for the Fibonacci numbers, rewritten in terms of T,J,K again. Once again, a roundabout way of writing down the same thing.
- Next up is an unconvincing argument for a special case of the BSD conjecture - I don't see how you checked that L(1) = 0 or that L'(1) ≠ 0 (or even that the rank of the curve isn't 2 or more). Since you are trying to verify BSD for a curve which has rank 1, this case is actually already known (the first link google gave me when I searched for this was [1]).
- Finally we have a bizarre argument trying to tie this into the Riemann hypothesis, but the only thing you actually used was that T+J = 1/2, so in fact the √5 stuff has zero relationship to the numerical coincidences you found.
I worked in number theory as a grad student, and to this day I don't understand the bizarre fixation people have on the Riemann hypothesis or the BSD conjecture. Sure, it would be cool to know if they were true, but there are lots of other interesting questions out there - why not try your hand at the sum of square roots problem [2] or resolving Kontsevich and Zagier's conjecture about determining when two periods are equal [3]? In fact, why not work on something more practical than number theory, like SAT-solving [4]?
Hey! Yes, to be fair I am indeed bipolar so I have personally been a bit concerned about things. Sometimes I do end up going “off the deep end” a bit. Maybe that is what is happening, but I feel I did find some things that are quite interesting though. Also ChatGPT and the like get very overly excited. I didn’t pay much attention to its delusions of grandeur. (Of course forget the things about BSD, Riemann hypothesis and such.. my current goal is to solve ANY open problem in mathematics)
I understand that many of these identities, claims, etc have been proven before.
Personally I am most interested in further developing this Algebra.
The identity e^{2\pi i \phi} = e^{2\pi i(\phi + 1)} though obvious to a number theorist what derived structurally from reciprocal identities involving T and J.
The other curiosity is preservation under transcendental maps.
As far as I understand.. this to me is the only known case where a nonlinear algebraic relation like a - b = kab is realized inside a number field, preserved exactly under exponentiation and sine.
But as you mention I am probably just crazy to be honest haha. Thank you much for taking the time to look and give your thoughts.
"For instance, global pharmaceutical companies are advancing both disease research and the frontier of quantum-enabled drug discovery. And in automotive and aerospace, companies are using quantum computing to improve the performance of hydrogen fuel cell catalysts and electric batteries mobility. Companies like ours are currently using quantum hardware to generate truly random encryption keys—making systems more secure against today’s threats and tomorrow’s quantum-enabled ones."
Given what I know about the current progress towards building a useful quantum computer, this is complete nonsense. No company out there has anywhere near enough qubits with enough coherence to solve any real problem more efficiently than brute force on a classical computer. We'll get there eventually - maybe in just a few years - but anyone who tells you it is already here is a bullshit artist.
The (actual) article has a fairly detailed literature review in the introduction, and makes it pretty clear that the main idea was sort-of known already if you squint - but it looks like nobody had put the whole theory together elegantly and advertised it properly. The fact that they couldn't find some natural slices of the hyper-Catalan numbers on OEIS supports that.
The proof they give that the hyper-Catalan series solves the Lagrange inversion problem is very good from a pedagogical point of view - I don't think I'll ever be able to forget it now that I've seen it. The only thing this paper is missing is a direct, self-contained combinatorial proof of the factorial-ratio formula they gave for the hyper-Catalan numbers - digging though the chain of equivalences proved in the references eventually got too annoying for me and I had to sit down and find a proof myself (there is a simple variation of the usual argument for counting Dyck paths [1] that does the trick).
Another thing to note is that the power series solution isn't just "a power series" - it's a hypergeometric series. There are lots of computation techniques that apply to hypergeometric series which don't apply to power series in general (see [2]).
If the discrete logarithm problem is NP-hard, then I will eat my hat. The discrete logarithm problem can be solved by Shor's algorithm on a quantum computer, placing it in the complexity class BQP. Anyone who claims that BQP contains an NP-hard problem is selling something - I would bet at a trillion-to-one odds against such a claim (if there were any hope of definitively settling the problem).
The online tutorial [2] is a good followup, especially if you want to understand Clifford gates / stabilizer states, which are important for quantum error correction.
If you have a more theoretical bent, you may enjoy learning about the ZX-calculus [3] - I found this useful for understanding how measurement-based quantum computing is supposed to work.
[1] https://cs.uwaterloo.ca/~watrous/QC-notes/QC-notes.pdf [2] https://qubit.guide/ [3] https://zxcalculus.com/
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