All Questions

The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying $$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$ are absurdly high, namely $$(...

Given a Diophantine equation it is not decidable if it has integer solution. I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties every member in $\mathcal D_{unique}$ is a ...

According to Matiyasevich, the existence of integer solutions of systems of polynomial equations with integer coefficients is undecidable. By introducing additional variables substituting factors of ...

Let $O_k$ be the ring of integers in a subfield $k$ of $\overline{\mathbb{Q}}$. Let's call an equation $p(x_1, \dots, x_n) = 0$ where $p$ is a polynomial in $n$-variables $x_1, \dots, x_n$ with ...

I have a quadratic Diophantine equation system which is possibly not homogeneous and has some mixed and some linear terms. But I know that there are only finitely many solutions over the integers. One ...