Number Theory, easy research?

Today I was talking to a professor of mine who said that he regrets never publishing as an undergrad because when he first applied to get a graduate degree internationally he was asked how many papers had he published before and as his answer was 0 that killed him for such a competitive position and that stagnated his career by years. He then said that you should start writing and thinking about publishing since your junior year.

I am currently a sophomore and I want to have an easy graduate life. I don't want to have to worry about getting accepted or not, so obviously now I want to start researching by next year.

My question is, if I were to come up with really weird problems in number theory, like fucked up diophantine equations, and started solving them and publishing a paper out of them. Is that good enough undergraduate research?

I like number theory because in number theory you can make a problem out of nothing. Everything is a problem as long as you can word it using enough quantifiers and I want to milk that as much as I can. I just wonder if this will work.

Will it? Any number theorists who do this?

Go figure out how to n-sect a general angle in 2 dimensional space with a ruler and compass (and perhaps an ellipse drawing tool)

No, seriously.

And then teach me how to do it.

Oh, I forgot to ask. If not then what is the minimum that you would consider publishable?

I just want to know what is the bare minimum and then train to reach that bare minimum, maybe before I even become a Junior, and then I can get a head start publishing.

Isn't this impossible for n>2?

Also, this is algebra not number theory. I need number theory because I already started reading number theory at a higher level than I am at so I can't turn back on that.

If I am smart enough I could learn graduate level number theory by my junior year. But I can't do the same for other fields like Algebra. I already picked my poison.

It's impossible to do in a finite number of steps, according to all known methods using ruler and compass
But there are ways to trisect an angle that don't involve ruler and compass
If you figure out how to 5-sect or 7-sect it would be the holy grail of hardcore Euclidean geometry

And I fail to see the distinction between number theory and geometry but perhaps that's because I've been doing geometry for too long.

>And I fail to see the distinction between number theory and geometry but perhaps that's because I've been doing geometry for too long.

Number Theory is like:

>Is this set of numbers finite or infinite? And if it is infinite then how dense? Denser than the primes?

>Does an integral solution for this equation exist? Do infinitely many or finitely many exist? How can they be classified?

>Does an integral solution for this class of equation always exist? Sometimes? For infinitely many kinds or finitely many? In the ones where solutions exist, are they infinite or finite?

What's the point in solving diophantine equations

just rape them with a gay men GPU and publish your solutions

Undergrad research is by no means necessary, and I'm fairly surprised a prof told you this. For the majority of math students do not publish in undergrad, schools do not expect publications from you, and there's a reason for that. The prerequisites are way too high.

You can probably do something if you really want but understand that you're basically limited to borderline trivial crap, or doing an applied project that is somewhat less crappy. Unless you get astronomically lucky.

>What's the point in solving diophantine equations
>just rape them with a gay men GPU and publish your solutions

Well, the idea would be to find some kinda complicated but deceptively simple family of equations and then proving that solutions exist, maybe prove that a certain algorithm solves them all, etc.

Not just solve them.

To be completely honest, the professor who told me this is a physics major. Maybe that is why? Are physics people expected to do research?

That said, even if I don't have to I'd rather do it. I feel like it would push me my limits. I already plan to attempt to study Analytic Number Theory to beyond the undergrad level and I would like to use that in some way or another. Even if it is trivial I am sure having my name somewhere.

Furthermore, to graduate you need to write a senior thesis so either way I have to write something. The deal is getting it published in a journal and/or doing more than one project.

you should really be talking to your professor about this, not shitposters on Veeky Forums

There is a lot of combinatorics research that has low hanging fruit which a dedicated undergraduate may be able to contribute by working hard.

Physics majors are not expected to publish in undergrad. However if you get the chance to publish it obviously helps.

t. Physics undergraduate thinking about grad school

To be quite honest I am a little afraid of looking ridiculous.

I have a feeling that my number theory professor thinks I am retarded, even though I am not because I am the only one in his class who can follow him up. Every time he asks for how to do the proof he is working on the board I am the only one giving suggestions and most of the time they work but when they don't work he looks at me straight in the eye and tells
>... Thank you...
with a condescending voice.

I can't trust that guy.

i know math professors can seem kind of unapproachable sometimes, but you're not going to get anywhere if you do everything on your own. you need someone who knows the field and can suggest problems that are appropriately difficult for you while still being somewhat interesting (i.e. publishable) to the wider community. not to mention if you want to go to math grad school, you'll need letters of recommendation, and you'll have to interact with people like your professor on a daily basis, so you might as well start now.

There is so many proofs that this is impossible with just a ruler and compass.

That said though I'm not one hundred percent sure about the 7-sected angle because it's a fermat prime.

Also any length can be constructed as long as you can formulate it in terms of addition, multiplication, addition, division and square roots.

>and can suggest problems that are appropriately difficult for you while still being somewhat interesting (i.e. publishable) to the wider community.

I can find someone else to do this. There is another number theory PhD who would have taught this class but this year he got involved in some huge research project and he got a special position and this year he won't be giving any classes.

Sad but I can live with that.

And for letters of recommendations, all of my freshman year professors love me. I am sure that I can get letters if I need them.

That said, the semester is still young so I'll see if I can make my number theory professor like me more.

Prove that for all integers n>2 there exists integers a and b which satisfy the equation in pic related.

I'll do it if this is an open problem.

If it just some textbook math puzzle I'd rather not as I have more important problems to work on.

Post proof of it being unsolved and I'll do it my main man.

That's the goldbach conjecture you fool.

I had never seen it expressed that way.

That is honestly too hard for me. Maybe after I get through analytic number theory I could try throwing a few jabs at it for fun.

Its never expressed like that, it's just another representation I made to shitpost on sci.

Fuck you my main man.

you're right; the number theory could probably do that too, and would be a good guy to know. but for the letters, you need someone who can vouch for your research ability, since that's primarily what the admissions are based on. a letter that says "user took my class his freshman year and got an A" won't do you that much good. i'll bet your worries about this professor are overblown.

Sorry my man.
How about this one.
Prove that Q≠NQ
Q represents all the problems which can be solved by a computer in polynomial time and NQ represents all the problems which cannot be solved in polynomial time.

here's one for you. prove that 3x+1=4 where x is the truth value (1 or 0) of the riemann hypothesis

You can only n-sect any angle in d dimensions iff $\frac{n}{d^m} = 1$, $m\in\mathbb{N}$

ffs how you tex on Veeky Forums?

Like this
[math] \frac{n}{d^m} = 1, m\in\mathbb{N} [/math]