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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.

MathPassenger
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    What do you think - is there for each $n\ge 1$ a $q<10^n$ such that $10^n+q$ is prime? If yes, then we have infinitely many such primes, of course. – Dietrich Burde Oct 01 '22 at 14:24
  • Primes in Twitter arithmetic progression... – Bob Dobbs Oct 01 '22 at 14:46
  • Related posts from mathoverflow that show that your question is hard, but confirm your intuition about such primes being rare (and in finite number for at least some values of $q$): https://mathoverflow.net/questions/5323 and https://mathoverflow.net/questions/337598 – charmd Oct 01 '22 at 14:59
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    Also, note that this post (https://math.stackexchange.com/questions/4523193/) about "usual unanswerable questions about primes on math.SE" mentions among these: "Are there infinitely many primes of the form $2^k + a$? Or any exponential expression". Since this post asks this exact question with $10$ instead of $2$, I assume it is not answerable to this day either – charmd Oct 01 '22 at 15:05
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    Well, except for $q\in{2,,5}$. Here's a heuristic but non-rigorous argument, which buoys but doesn't prove your intuition: the prime number theorem says in a certain sense a large natural number $N$ has probability $1/\ln N$ of being prime, which for $N=10^n+q$ is asymptotic to $1/(n\ln10)$. Since this series diverges under $\sum_n$, for each prime $q$ not dividing $10$ we expect infinitely many $n$ to make $N$ prime. – J.G. Oct 01 '22 at 17:26
  • Probably, there are already infinite many primes of the form $10^n+3$. However , I have doubts that this much weaker conjecture can be proven. We can make a nice project of this question (maybe, someone has done this already!) : For $n=1,2,\cdots$ upto some reasonable limit , search the smallest prime $q$ , such that $10^n+q$ is prime as well. The existence of such a prime is implied by Dickson's conjecture. – Peter Oct 03 '22 at 11:40

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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.

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