MathOverflow

From our note with the Claude AI Congruences for the matrix permanent mod $3,4,6$
Let $M$ be $n \times n$ symmetric integer matrixwith all entries on the diagonal even.We have $\operatorname{per}(M) \equiv \det(M) \pmod 4$ if$n\not\equiv2 \pmod 4$ and $\operatorname{per}(M) \equiv -\det(M) \pmod 4$ if $n \equiv 2 \pmod 4$....

The question is to find all integer solutions to the equation$$y^2+z^2=x^3+1.$$This equation obviously has infinitely many integer solutions (take, for example, $(x,y,z)=(u^2,u^3,1)$ for any integer $u$) but the question is to describe all integer solutions...

It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the topology.
For $\mathbb{R}$, this is obtained as follows: the condition $x\ge 0$ is equivalent to $\exists y:y^2=x$, hence the ordering on $\mathbb{R}$ is preserved by any field automorphism. Since such an automorphism is the identity on $\mathbb{Q}$, it must be...

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof...