March 9th, 2012, 01:39 AM  #1 
Member Joined: Feb 2012 Posts: 38 Thanks: 0  primes
Hi. Let and be primes different of and . Show that if is a power of two integers then is divisible by Thanks. 
March 9th, 2012, 06:50 AM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: primes
Well, if both p and q are not equal to either 2 or 3, then both and are congruent to 1 mod 3, which means is divisible by 3. But , so it remains to show that is not divisible by 3. I'm not quite sure what you mean when you say that is a power of two integers.

March 11th, 2012, 09:52 AM  #3 
Member Joined: Feb 2012 Posts: 38 Thanks: 0  Re: primes
Thanks, but I'm not using congruences yet.

March 11th, 2012, 07:35 PM  #4  
Member Joined: Feb 2012 Posts: 38 Thanks: 0  Re: primes Quote:
Thanks  
March 12th, 2012, 07:23 AM  #5 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: primes
The conjecture is false. Consider the example . , which is a power of two integers, but , which is not divisible by 3. Another example is . , which is a power of two integers, but , which is not divisible by 3.

March 12th, 2012, 08:22 AM  #6 
Member Joined: Feb 2012 Posts: 38 Thanks: 0  Re: primes
Well, I've wrote the exercise more logically. this is: If and are primes differents of and and then I hope it'll be correct. Thanks 
March 12th, 2012, 08:36 AM  #7 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: primes
Ok, I can solve this using congruences. Neither p nor q is divisible by 3. Since , (mod 3) or (mod 3). If it is the former, then p = 2 mod 3 and q = 1 mod 3. If it is the latter, then p = 1 mod 3 and q = 2 mod 3. In either case, (mod 3); i.e., it is divisible by 3. I have an interesting exercise: Prove that if p and q are both primes different from 2 and 3, exactly one of and is divisible by 3. 

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