I found a very interesting problem in the [Open Problem Garden](http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping), which I am surprised is not as well-known as I would think it would be:


Prove that If $p>3$ is prime and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb Z\_p}$, then there are column vectors $x,y \in {\mathbb Z\_p}^n$ which have no coordinates equal to zero such that $Ax=y$.


It is easy to show that this conjecture is false when $p=2,3$. Also, when $p$ is a little larger than $n$, this conjecture has been [proven true](https://doi.org/10.1007/BF01215347). However, the case when $n$ is large relative to $p$ seems not only difficult to prove, but also it seems to require nontrivial computational resources to even get a sense of what is going on.


Has any work been done to try to computationally understand this problem? In particular, is this conjecture known to be true for say $p=5$, $n=15$?



A nowhere-zero point in a linear mapping conjecture
This is the Alon-Jaeger-Tarsi conjecture first stated in 1981 and [resolved](https://arxiv.org/pdf/2107.03956.pdf) very recently (for $p\ge 83$) by János Nagy and Péter Pál Pach.


