Prime $8263\equiv 1\pmod {17}$ and $8\cdot 2\cdot 6\cdot 3\equiv -1\pmod {17^2}$.
Are there other odd primes $p$ without digit $0$ such that:
$p\equiv 1\pmod q$ and the product of the digits of $p$ is $\equiv -1\pmod {q^2}$?
with $q$ some other odd prime? In particualar I ask for other solutions for $q=17$.
Yes, the next few examples for $q=17$ are 21863, 42433, 49811, 92821, 118661, 138143, 143243, 182921, 198221, 223381...
There are plenty of smaller solutions for some other $q$, mostly with $q=3$ (the first such example is 157) but also a few with $q=5$ or $q=7$; additionally 1873 works for $q=13$ and 3851 for $q=11$.