A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it. This is the actual question which I'm tr …

Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at http://mathoverflow.net/questions/131353/help-with-this-sys …

A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^ …

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\x …

Could you state or direct me to results regarding the Diophantine equation $a^2+b^2=c^2+d^2$ over integers? Specifically, I am looking for a complete parametrization. In the case t …

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems tha …

It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The …

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a c …

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm …

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken. I felt …

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial t …

I came across the following simple question: what odd integer squares have exactly 3 ones in their binary expansion? After looking at it for a while I convinced myself that the on …

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known. (1) has infinitely many solutions in an extension of $\mathbb{Z}$ (root of $9-15x+37x^2 $ ) resultin …

Dear all, I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions. The equation $x^n …

Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers? For example, if there are in …