What if I asked you to find me two non-integer rational numbers [math] a,b [/math] such that all of [math] a-b, a^2 - b^2, a^3 - b^3, a^4 - b^4 [/math] are integers? Would you be able to do it? What if I then added that the denominator of [math] a [/math] must be exactly 123456789?
How did I come up with these numbers? Was it a super computer I had running for 10 weeks before making this thread? WRONG! It is the application of a theorem I proved. A theorem that can be used to find rational numbers with the property I just outlined. In fact, it can be used to find all of them! And it can even be fine tuned to find ones with specific properties, like the ones I just found for you. Hopefully you like rational numbers as much as I do. And if you do then you should read my paper. All that is needed to understand the proofs is elementary number theory: the GCD, congruences, the chinese remainder theorem, and the manipulation of the floor/ceiling functions. And if you lack these techniques but you would like to apply my theorems then to understand the final results (which are formulas to generate these special rational numbers) then you only need to understand modern mathematical notation.
It is 9 pages long (the last page is only the references) and it contains four theorems. The first theorem is about existence, and the other three theorems are constructive characterizations of these rational numbers. Pic related is a diagram in the paper.
How can you turn this knowledge into income into wealth?
Asher Martinez
Get Fields medal, proceed to instant tenure.
Angel Parker
>Why are you doing this?
Good question. As I outline in the "motivation" section of my paper, this week I was made aware of the following property of real numbers:
If [math] a,b \in \mathbb{R}, a \neq b [/math] with [math] a - b, a^2 - b^2, a^3 - b^3 , ... [/math] all integers then necessarily both a and b must be integers.
Then I immediately asked myself what would happen if instead of going to infinity, I stopped at some number [math] n [/math]. Perhaps just go up to [math] n=3 [/math] or [math] n=4 [/math]. Could I find non-integer real numbers that satisfy the property? With a little tinkering I found that indeed it was the case (took me two days to find my first example for n=4).
Then after I explored this concept and made many tables for formulas to generate these (these formulas were special cases of the theorems I present in the paper. My paper has the most general solutions). I finally saw the light. I saw the patterns. With some ingenuity and outright abuse of the chinese remainder theorem I finally had a full characterization of the solutions. And it was beautiful.
This took me about 4 days. I was able to do so much work so quickly because I was away on vacation so I basically had 24 hours free time to do mathematics. Then, just yesterday I came back home and with energy in my hands and my heart I decided to write all of my notes concisely and condense them into the four theorems you see before you.
>How can you turn this knowledge into income into wealth?
I don't know. I plan to show this to my number theory professor if no one here finds a mistake and tells me I'm a brainlet and perhaps then my professor will like me and invite me to Japan to study inter universal teichmuller theory or something who knows.
For now I just want to get feedback on my method and my theorems. Vixra allows for modifications so if there is anything I can improve before I show it to my professors IRL then I will do it.
Luis Wright
How do the people who pay for Fields medal get wealth from this?
How do the people who pay your salary in your tenure get wealth from this?
I understand the issues with Vixra, but I have no academic connections (currently undergrad) and arxiv's rules say that papers must be relevant in the modern advances of the field, which means that elementary number theory papers would not be allowed.
If vixra allows me to put my papers online so that I can then post them here then that is enough for me.
Connor Collins
Christ user, if you're an undergrad show this to one of your professors and ask him to endorse you on arxiv. You can also submit it to a journal. You screw yourself over by putting it up on vixra. I don't even know if it's correct, since my first instinct was to wince at the site you used. The vast majority of mathematicians will dismiss your shit out of hand the same way. Vixra is for cranks and loons.
Caleb Gray
I think you are being over dramatic. I just posted there to be able to get feedback, and vixra allows you to delete your paper if you want (though I don't really know if that is necessary).
My paper is not really important, it is by all means recreational mathematics. I think that no harm is done by publishing it there. The results speak for themselves.
