My background on number theory is very weak, so please bear with me...


Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\mathbb{F}\_p^{n\times n}$ are similar to each other. Does this yield the existence of a matrix $X\in\mathrm{SL}\_n\left(\mathbb{Z}\right)$ satisfying $AX=XB$ ? What if we additionally assume $A$ and $B$ to be similar in $\mathbb{Q}^{n\times n}$ ?



Local-globalism for similar matrices?
Your question reminds me of a classical theorem of Latimer and MacDuffee (Annals of Math. 1933). To be sure, the theorem does not answer your question, but it seems relevant.


A nice contemporary treatment of this theorem (with references) can be found at 


<http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/matrixconj.pdf>


[At the moment that I write this, math.uconn.edu seems to be down. I presume this is only temporary!]


