My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 14th, 2009, 05:48 AM   #1
Member
 
Joined: Jul 2009

Posts: 55
Thanks: 0

Odd primes as n+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
Euzenius is offline  
 
October 14th, 2009, 06:10 AM   #2
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Odd primes as n+2

Quote:
Originally Posted by Euzenius
MY QUESTION
Why they have to be an odd multiple of 3 ? Is it only a conjecture ?


Euzenius
My guess is:
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.
cknapp is offline  
October 14th, 2009, 11:14 AM   #3
Member
 
Joined: Jul 2009

Posts: 55
Thanks: 0

Re: Odd primes as n+2

Well done !

Euzenius
Euzenius is offline  
October 14th, 2009, 12:14 PM   #4
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Odd primes as n+2

Quote:
Originally Posted by Euzenius
Well done !
Thanks. My very little bit of experience with number theory problems suggests "primes of the form..." are always stated to have "known" constraints which have never-seen proofs. They tend to be easy to prove if you see the trick, but the trick is sometimes a bit hidden...
cknapp is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
odd, primes



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
primes and twin primes: Number between powers of 10 caters Number Theory 67 March 19th, 2014 04:32 PM
primes PerAA Number Theory 4 October 18th, 2013 08:25 AM
Primes in Z ? gelatine1 Algebra 4 September 1st, 2013 10:09 PM
n^2+1 n^2+n+1 primes johnr Number Theory 20 April 9th, 2013 05:49 PM
primes agustin975 Number Theory 11 March 10th, 2013 05:40 AM





Copyright © 2019 My Math Forum. All rights reserved.