Primes of the form $10^n + q$. Nervous for the Chelsea game coming up in a bit, I went on Twitter, and the Twitter account @_primes_ mentioned that the number $1000003$ is a prime. This got me thinking: given prime $q$ and a natural number $n$ with $10^n \gg q$, how many primes of the form $10^n + q$ are there?
I feel as if someone must have asked this question in the past—I presume that there are infinitely many such primes—but the best I could do was to show that it is not always true that $10^n + q$ is prime if $q$ is prime (take $q = 5$ for instance, and choose any $n$). But I cannot prove this, and I was curious about whether anyone has any proofs or resources about this.

See Lemma 2 in Sury, B. "Extending Given Digits to make
Primes or Perfect Powers". Resonance (Oct 2021). pp. 941-947. URL: https://www.ias.ac.in/article/fulltext/reso/015/10/0941-0947

Lemma 2. Let $A$ be any given natural number. Then, one may add digits
to the right end of the digits of $A$ to obtain a prime number.

This however does not take into consideration the constraint $10^q ≫ n$ or that $q$ is prime. But, you should be able to build upon that.
