Let $p_1,p_2$ be primes and $x\in\mathbb{N}$. I want to investigate \begin{equation*} p_1^x\equiv p_2^x\equiv 1 \pmod{p_1p_2-1} \end{equation*} I want to find how $x$ depends on $p_1$ and $p_2$.
This is clearly something to do with the order of the primes but I can't find anything on comparing primes with the same order.
One way I thought to do this was to look at it as \begin{equation*} p_1^x-1\equiv p_2^x-1\equiv 0\pmod{p_1p_2-1} \end{equation*} and look at how their cyclotomic polynomials interact with each other, but cyclotomic polynomials don't really have any results for this.
For specific values this is easily evaluated. For example; $p_1=7$ and $p_2=11$, we get
\begin{equation*} 7^6=117649\equiv11^6\equiv1771561\equiv1\pmod{76} \end{equation*} so here $x=6$.
Any advice would be very appreciated.
$\!\!\bmod p_1 p_2\! -\!1\!:\,\ p_1 p_2\equiv 1\Rightarrow p_2\equiv p_1^{-1}\,$ so $\,p_1^n\!\equiv\! 1\iff p_2^n\equiv 1,\,$ so your question boils down to finding the order of $p_1$ (= order of $p_2),\,$ necessarily a divisor of every $n$ such that $\,p_1^n\equiv 1\,$ (e.g. the order must divide the Euler phi and Carmichael lambda function of the modulus).