Hello! I would love for someone from the mathematical community to take a look at this. Curious what people think. :)
cevi•8mo ago
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.
tristenharr•8mo ago
cevi•8mo ago
- 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]?
[1] https://mathoverflow.net/questions/309086/bsd-conjecture-for... [2] https://en.wikipedia.org/wiki/Square-root_sum_problem [3] https://en.wikipedia.org/wiki/Period_(algebraic_geometry)#Op... [4] https://satcompetition.github.io/
tristenharr•8mo ago
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.