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I have recently been learning about cubic characters, and the machinery of Gauss and Jacobi sums used to prove the cubic reciprocity theorem, and using this, I can now determine when any prime is a cu …

I have been looking into the Fermat quintic equation $a^5+b^5+c^5+d^5=0$. To exclude the trivial cases (e.g. $c=-a,d=-b$), I will take $a+b+c+d$ to be nonzero for the rest of the question. It can be s …

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$ $ $ $ $ $ $ $ $ $ $ $d^2$ $e^2$ $f^2$ $g^2$ Wh …

I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the bas …

I have been studying the equation $a^6+b^6+c^6=d^2$, trying to find rational solutions. I know it is a K3 surface, with high Picard rank, so there should be rational or elliptic curves on it. When Elk …

I was considering the Fermat Catalan conjecture, where the equation $a^m+b^n=c^k$ has only finitely many nontrivial solutions (with coprime $a, b, c$) with $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}=1$ (and …

In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is …

If we rewrite the equation as $y^2+t^{2}z^3 = x^3$, and let $v=tz$, then $y^2+zv^2=x^3$. If you factorize over $Q[\sqrt{-z}]$, then $(y+v\sqrt{-z})(y-v\sqrt{-z})=x^3 $. Let $x=(a+b\sqrt{-z})(a-b\sqrt{ …

This is a similar question to https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487 Let f(n) be the maximum n …

I have read, from the question Irreducibility of polynomials in two variables, that all polynomials $f(x)-g(y)$, where $f, g$ are indecomposable polynomials, and there are no $a, b$ such that $g(ax+b …

I have been working on solutions to $x^5+y^5+z^5=1$, and I found that the three solutions of $x^3+bx+\frac{1}{5b}$ satisfy that equation. Multiplying by $5b$: $5xb^2+5x^3b+1=0$, then solving for b yie …

I found a parameterized solution of $X^5+Y^5+Z^5=T^5$ with $X$ and $T$ rational and $Y$ and $Z$ complex rational: $(k^2-4k+1)^5+(2k-2+(k^2-2k+3)i)^5+(2k-2-(k^2-2k+3)i)^5=(k^2-3)^5$ There is an almos …

I was looking into cyclotomic extensions of the natural numbers, and I found that extending the naturals with the 23rd root of unity caused the ring to no longer be a UFD. In other words, there now ex …

I recently read http://cp4space.wordpress.com/2013/11/25/crash-course-in-gaussian-integers/, where it teaches you how to solve some diophantine equations of the form x^2+n=y^3 using the gaussian integ …