20200304, 18:28  #1 
Mar 2018
1013_{8} Posts 
Numbers of the form 667, 666999667, 666999666999667...
When these numbers are prime are they always congruent to 15 mod 23?
666999666999667 is prime and 15 mod 23 Last fiddled with by enzocreti on 20200304 at 21:02 
20200305, 02:51  #2 
Romulan Interpreter
Jun 2011
Thailand
2×11×397 Posts 
The answer is NO.
If you index them from 0, (i.e. your prime is the index 2), then index 13 is also prime (13*6+3 digits) and 15 (mod 23), which is for sure just coincidental. There is no other prime till index 1000 (6003 digits). This is quite fast to test with Pari, it takes about 510 minutes, and I assume you already knew both things, and deliberately omit the index 13 in the post, just for the sake of trolling. You expected some silly guy (like me ) to jump and test, and find index 13 is prime, and find index 13 is 15 (mod 23), and don't find any other prime, and answer "yes" to your question, but theoretically, there is absolutely no reason why there should be no other primes in the series, which are not 15 (mod 23), especially as 23 is a prime, and the residues mod 23 repeats in a pattern (0,12,15,10,3,7,8,14,4,13,21) (WHY?? Think about). They are "sievable" in the sense that every index 7k+4 is divisible by 7, every 11k+6 is divisible by 11, every 13k+7 is divisible by 13, every 8k+3 is divisible by 17, every 11th is divisible by 23, and so on. What remains after sieving "might" be prime, and albeit they are quite rare, there is no reason for them to be always the index 11k+2 in the string (i.e. 15 (mod 23)). These kind of "series" are quite easy to construct, you pick some arbitrary format, do some modular tests, etc, at the end, they mean nothing, beside of pissing some other people off. I believe this is in fact your intention. You should quit wasting your time (and ours) and start reading some math. Nick's topic about number theory on this forum is an excellent starting point. Last fiddled with by LaurV on 20200305 at 02:59 
20200305, 02:57  #3  
"Curtis"
Feb 2005
Riverside, CA
2×5×431 Posts 
Quote:
I think it's time we delete these questions until Enzo shows any effort at all to do his own work. Dozens of threads with "I found one of these, are there more?" is a waste of forum space, never mind the time of mods to read this crap. 

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