For a given positive integer $k$ , the task is to find a prime number of the form $$2023^n+k$$ with positive integer $n$ , or to show that there cannot be such a prime.
I am particular interested in the case $k=222$ because for all smaller positive integers $k$ , there is either a prime of this form or small prime factors are forced.
However, there are positive integers $n$ such that $$2023^n+222$$ has no small prime factor (for example for $n=47$ , we have a semiprime and the smaller prime number has $17$ digits) , but there is no positive integer $n\le 13\ 000$ such that $$2023^n+222$$ is prime.
Is there a prime number of this form ?
Best Answer
You stopped just a little too early, $n=17315$ results in the PRP $2023^{17315}+222$ (with 57244 digits) to all prime bases less than $100$ giving a large probability of primality.
I did not find any other prime/PRP of this form with $n < 20\ 000$ and $k=222$.