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 March 9th, 2012, 01:39 AM #1 Member   Joined: Feb 2012 Posts: 38 Thanks: 0 primes Hi. Let $p$ and $q$ be primes different of $2$ and $3$. Show that if $p-q$ is a power of two integers then $p+q$ is divisible by $3$ 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 $p^2$ and $q^2$ are congruent to 1 mod 3, which means $p^2 - q^2$ is divisible by 3. But $p^2 - q^2= (p - q)(p + q)$, so it remains to show that $p - q$ is not divisible by 3. I'm not quite sure what you mean when you say that $p - q$ 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
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Quote:
 Originally Posted by icemanfan I'm not quite sure what you mean when you say that $p - q$ is a power of two integers.
Mmmm for example $11-7=4=2^2$ then $3|(11+7)=18$, which it's true.

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 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 $p= 73, q = 37$. $73 - 37= 6^2$, which is a power of two integers, but $73 + 37= 110$, which is not divisible by 3. Another example is $p= 293, q = 149$. $293 - 149= 12^2$, which is a power of two integers, but $293 + 149= 442$, 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 $p$ and $q$ are primes differents of $2$ and $3$ and $p-q=2^n$ then $3|(p+q)$ 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 $p - q= 2^n$, $p - q= 1$ (mod 3) or $p - q= 2$ (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, $p + q= 0$ (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 $p - q$ and $p + q$ is divisible by 3.

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