The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$.


I am a prospective undergraduate mathematics student in Zimbabwe and I recently obtained a new result that the above equation indeed has no further integer solutions such that $ \frac{1}{3} (x + y + z)$ is a perfect square.


I also came up with a new and surprisingly elegant elementary proof of the Erdos-Moser conjecture for odd exponents that the Diophantine equation:


$1^n + 2^n + \cdots + m^n = (m+1)^n$ has no integer solutions other than the trivial solution $1+2=3$ if $n$ is odd. 


So which journal is best for me to submit my findings ? I was thinking of the Journal of Number Theory ? Also, i'm intending to apply for undergraduate admission in U.S.A / U.K, how can these results strengthen my admission prospects ? And which universities are most interested (or most suitable for) in undergraduate applicants who already have done some research before. ? 



A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$
My opinion is:  

**Step 1**.  

Repeat the arguments you use in your proofs again and again.  

If you are still sure that you have some new (valid) things to say about the equation then go to  

**Step 2**.  

Learn how to use [LaTeX](http://arxiv.org/) ,write your paper and submit it to the [arXiv](http://arxiv.org/).  

If it is submitted, then go to  

**Step 3**.  

Choose a journal which has previously published papers related to the equation you mention and submit it for publication.  

(Don't choose the best journal but a serious one) 


It would be better if you choose a title which won't make the editor think that he will probably waste the time of the referees.  

A title "Some notes on the equation $x^3 +y^3+z^3=3$" will be safer. 


Good luck!!!


