October 14th, 2009, 05:48 AM  #1 
Member Joined: Jul 2009 Posts: 55 Thanks: 0  Odd primes as n²+2
Hello everybody Sloane A056899 says 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891 COMMENT Note that all terms after the first are equal to 11 modulo 72 and that (a(n)11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime and if k=6m+3 then a(n)=k^2+2=72m(m+1)/2+11. MY QUESTION Why they have to be an odd multiple of 3 ? Is it only a conjecture ? Euzenius 
October 14th, 2009, 06:10 AM  #2  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Odd primes as n²+2 Quote:
If n is not divisible by 3, then n^2 = 1 mod 3, so 3  n^2+2. So n must be divisible by 3 for n^2+2 to be prime this also means they are 2 more than an odd multiple of 9. They have to be two more than an odd multiple, because if they were 2 more than an even multiple, they would be even.  
October 14th, 2009, 11:14 AM  #3 
Member Joined: Jul 2009 Posts: 55 Thanks: 0  Re: Odd primes as n²+2
Well done ! Euzenius 
October 14th, 2009, 12:14 PM  #4  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Odd primes as n²+2 Quote:
 

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