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 2019-04-15, 16:50 #1 enzocreti   Mar 2018 523 Posts Prime 5101 The prime 5101 is congruent to 3^6 (mod 1093) Are there infinitely many primes p congruent to 3^k (mod 1093) with k>0?
2019-04-15, 17:01   #2
CRGreathouse

Aug 2006

31·191 Posts

Quote:
 Originally Posted by enzocreti Are there infinitely many primes p congruent to 3^k (mod 1093) with k>0?
For any positive integer k there are infinitely many primes congruent to 3^k mod 1093, yes. Note that 3 is relatively prime to 1093 (and that 3^k has order 7 mod 1093).

Last fiddled with by CRGreathouse on 2019-04-15 at 17:02

2019-04-15, 17:16   #3
enzocreti

Mar 2018

523 Posts
5101

Quote:
 Originally Posted by CRGreathouse For any positive integer k there are infinitely many primes congruent to 3^k mod 1093, yes. Note that 3 is relatively prime to 1093 (and that 3^k has order 7 mod 1093).

51001 divides 2^(5101-1)-1.

Just a curio

But maybe a better curio is that 1021 and 1201 divide 2^(5101-1)-1 where 1021 and 1201 are primes with 0 and 2 changed....1021+1201 is 2222...moreover 1801 and 8101 divide 2^5100-1...also here 8 and 1 changed

Last fiddled with by enzocreti on 2019-04-15 at 17:52

 2019-04-15, 18:14 #4 enzocreti   Mar 2018 523 Posts Amazing that... (2^5100-1)/1021/1201/1801/8101 is congruent to 9393 mod (101^2) (2^5100-1) is divisible by at least four primes of the form x^2+45*y^2 Last fiddled with by enzocreti on 2019-04-15 at 18:28

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