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As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple

• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033
• 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033
• 10000000000000000000000000000000003000000033
• 10000000000000000000000000000030000000000033
• 10000000000000000000000030000000000000000033
• 10000000003000000000000000000000000000000033

There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple

• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033
• 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033
• 10000000000000000000000000000000003000000033
• 10000000000000000000000000000030000000000033
• 10000000000000000000000030000000000000000033
• 10000000003000000000000000000000000000000033

There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.

Edit 4: As requested here are the sage scripts I used. To check if $10^n+333$ was prime:

for n in range(0,500):
  k=10^n+333
  if(is_prime(k)):
    print n

And to check for numbers of the form $10^n+3*10^i+33$:

for n in range(0,500):
  k=10^n+33
  for i in range(2,n):
    l=k+3*10^i
    if(is_prime(l)):
      print l

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033 10000000000000000000000000000000003000000033 10000000000000000000000000000030000000000033 10000000000000000000000030000000000000000033 10000000003000000000000000000000000000000033

There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple

• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033
• 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033
• 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033
• 10000000000000000000000000000000003000000033
• 10000000000000000000000000000030000000000033
• 10000000000000000000000030000000000000000033
• 10000000003000000000000000000000000000000033

There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

I'm currently checking between 2000 and 3000. I'll try to write something in a bit to check the various combinations of $10^n+3*10^i+3*10^k+3$.

Edit: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form i previously mentioned and indeed there are lots, here a few:

100000030000000000000030000000000000000000000003 100000000000000000000300000000000000000000003003 100000000000000000000300000000000030000000000003 100000000000000000300000000000000000000003000003 100000000000000000300000000000000000000030000003 100000000000000003000000000000000000003000000003 100000000000000030000000000000000000300000000003 100000000000000030000003000000000000000000000003 100000030000000300000000000000000000000000000003 100000000000006000000000000000000000000000000003 100000000003030000000000000000000000000000000003 100000000000300000000000000000000000000030000003 100000000000300000000000000000000030000000000003 100000000003000000000000000000000000000000003003 100000000003030000000000000000000000000000000003 100000000303000000000000000000000000000000000003 100000000030000000000030000000000000000000000003 100000000300000000000000000000000000030000000003 100000000303000000000000000000000000000000000003

In particular I was always able to find a prime number of that form for n in the range [0,231]. I accidentally stopped my search at this range. I could probably run something over night, but it seems safe to say that even if there aren't an infinite number, there are a lot of them.

As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

Edit 3: I stopped my laptop's search between 2000 and 3000, since it hadn't found anything in 20 minutes. I wrote a quick program to check numbers of the form $10^n+3*10^i+33$. Here are a couple

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300033 100000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000033 100000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000033 100000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000033 10000000000000000000000000000000003000000033 10000000000000000000000000000030000000000033 10000000000000000000000030000000000000000033 10000000003000000000000000000000000000000033

There seemed to be plenty of numbers of this form and presumably I could find more if I checked some of the other possible forms as outlined by dr jimbob.

Note: I revised the post a bit after jimbob pointed out I was actually looking for primes that didn't quite fit the requirements.