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March 9th, 2012, 01:39 AM   #1
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primes

Hi.

Let and be primes different of and . Show that if is a power of two integers then is divisible by

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March 9th, 2012, 06:50 AM   #2
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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.
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March 11th, 2012, 09:52 AM   #3
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Re: primes

Thanks, but I'm not using congruences yet.
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March 11th, 2012, 07:35 PM   #4
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Re: primes

Quote:
Originally Posted by icemanfan
I'm not quite sure what you mean when you say that is a power of two integers.
Mmmm for example then , which it's true.

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March 12th, 2012, 07:23 AM   #5
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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.
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March 12th, 2012, 08:22 AM   #6
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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.

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March 12th, 2012, 08:36 AM   #7
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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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