I saw it as a sort of science-fiction - imagine living in a world where the smartest intellectuals all struggled to solve basic exercises about graph theory. Really imagine living in such a world - would you not feel frustrated when you tried to explain this basic concept to these supposed experts and they just didn't get it? The main character must have felt like he was going crazy!
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.
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]?
"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).
There are plenty of mathematicians - mostly set theorists - who are actively working on finding new axioms of mathematics to resolve questions which can't be resolved by ZFC. Projective Determinacy is probably the most important example of a new axiom of mathematics which goes far beyond what can be proved in ZFC, but which has become widely accepted by the experts. (See [1] for some discussion about the arguments in favor of projective determinacy, and [2] for a taste of Steel's position on the subject.)
I suggest reading Kanamori's book [3] if you want to learn more about this direction. (There are plenty of recent developments in the field which occured after the publication of that book - for an example of cutting edge research into new axioms, see the paper [4] mentioned in one of the answers to [5].)
If you are only interested in arithmetic consequences of the new axioms, and if you feel that consistency statements are not too interesting (even though they can be directly interpreted as statements about whether or not certain Turing machines halt), you should check out some of the research into Laver tables [6], [7], [8], [9]. Harvey Friedman has also put a lot of work into finding concrete connections between advanced set-theoretic axioms and more concrete arithmetic statements, for instance see [10].
The error rate of human mathematical work is not zero, but it does go down exponentially with the amount of time that the mathematician spends carefully thinking about the problem. Mistakes tend to be the result of typos, time pressure, or laziness - showing your work to others and having them check it over does help (it's one of the reasons we have peer review!), but is absolutely not necessary.
If the error rate is low enough - and by simply spending a constant factor more time finding and correcting errors, you can get it below one in a million - then you do get a virtuous feedback loop even without tying in a proof-checker. That's how humans have progressed, after all. While you are right to say that the proof-checker approach certainly is not trivial, it is currently much easier than you would expect - modern LLMs are surprisingly good at converting math written in English directly to math formalized in Lean.
I do think that LLMs will struggle to learn to catch their mistakes for a while. This is mostly because the art of catching mistakes on your own is not taught well (often it is not taught at all), and the data sets that modern LLMs train on probably contain very, very few examples of people applying this art.
A tangent: how do human mathematicians reliably manage to catch mistakes in proofs? Going line-by-line through a proof and checking that each line logically follows from the previous lines is what many people believe we do, but it is actually a method of last resort - something we only do if we are completely lost and have given up on concretely understanding what is going on. What we really do is build up a collection of concrete examples and counterexamples within a given subject, and watch how the steps of the proof play out in each of these test cases. This is why humans tend to become much less reliable at catching mistakes when they leave their field of expertise - they haven't yet built up the necessary library of examples to allow them to properly interact with the proofs, and must resort to reading line by line.
When you get good enough at mathematics, you can tell if your proofs are correct or not without asking a TA to grade them for you. Most mathematicians reach this level before they finish undergrad (a rare few reach it before they finish high school). While AI hasn't reached this level yet, there is no fundamental barrier stopping it from happening - and for now, researchers can use formal proof-checking software like Lean, Coq, or Isabelle to act as a grader.
(In principle, it should be also be possible to get good enough at philosophy to avoid devolving into a mess of incoherence while reasoning about concepts like "knowledge", "consciousness", and "morality". I suspect some humans have achieved that, but it seems rather difficult to tell...)
I prefer his explanation to most other explanations because he starts, right away, with an analogy to ordinary probabilities. It's easy to understand how linear algebra is related to probability (a random combination of two outcomes is described by linearly combining them), so the fact that we represent random states by vectors is not surprising at all. His explanation of the Dirac bra-ket notation is also extremely well executed. My only quibble is that he doesn't introduce density matrices (which in my mind are the correct way to understand quantum states) until halfway through the notes.
MIT's PRIMES program does exactly this - they give advanced high school students a mentor who picks out a problem, gets them up to speed on what is known, and then they work on the problem for a year and publish their results. It tends to work best with problems which have a computational aspect, so that the students can get some calculations done on the computer to get the ball rolling.
If the Hanoi pieces alternate in color, there is another very easy algorithm: always avoid putting two pieces of the same color directly on top of each other. I noticed this by accident while watching someone else solve a Hanoi puzzle with alternating colored pieces. (The person I was watching didn't know this trick either. I suspect the manufacturers of the alternatingly colored puzzle also didn't know about it. So where did the knowledge come from?)
Have you tried to read any of the literature on the Risch algorithm? If you haven't, you might want to get started by taking a look at the paper "Integration in Finite Terms" by Rosenlicht [1] and chasing down some of the references mentioned in [2].
Of course, in the real world we don't give up on integrals just because they can't be expressed in terms of elementary functions. Usually we also check if the result happens to be a hypergeometric function, such as a Bessel function. If you want to get started on understanding hypergeometric functions, maybe try reading [3] (as well as the tangentially related book "A = B" [4]).