Catalan's conjecture (Mihăilescu's theorem) states that the only case of two consecutive integer powers is given by the equation $3^2-2^3=1$. We can easily verify that it is not valid in some prime fields $\mathbb F_p$. For example, one has $$\begin{cases}3^3-4^2=1\pmod5\\5^3-3^5=4^4-5^5=1\pmod7\\8^3-6^7=10^7-3^3=1\pmod{11}\end{cases}$$
The problem I want to try is the following:
Prove or disprove that for all $\mathbb F_p$ with $p\gt 3$ there is an example of $x^a-y^b=1\pmod p$, with $xy\ne0$ other than the trivial one $3^2-2^3=1$