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Old Mar10-09, 06:29 PM                  #1
Carl140

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prime numbers problem

Hello,

I can't get this small contest problem. How do you solve this kind of problem?

Let p and q be prime numbers such that (p^2+q^2)/(p+q) is an integer.
Prove p=q.
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Old Mar10-09, 07:28 PM                  #2
John Creighto

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Re: prime numbers problem

I can't figure it out.

p^2+q^2=n1*(p+q)

p(p-n1)+q(q-n1)=0

(p-n1)=n2*(q-n1)

p-n1+n2 n1=n2 q
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Old Mar10-09, 07:47 PM                  #3
Hurkyl

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Re: prime numbers problem

What if r divides the denominator?
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Old Mar10-09, 07:54 PM                  #4
Carl140

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Re: prime numbers problem

What's r?
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Old Mar10-09, 08:15 PM                  #5
Hurkyl

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Re: prime numbers problem

Originally Posted by Carl140 View Post
What's r?
Some number that happens to divide the denominator.
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Old Mar10-09, 08:23 PM                  #6
Carl140

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Re: prime numbers problem

I still don't get it, sorry. Can you please explain a little bit more?
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Old Mar10-09, 09:28 PM                  #7
John Creighto

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Re: prime numbers problem

I think I have a solution but I won't post it without moderator approval.
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Old Mar11-09, 01:16 AM                  #8
John Creighto

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Re: prime numbers problem

Okay, here is my hint. What theorem might be helpful to show what integer values of p+q will satisfy the following equation?

(p+q)^2-m(p+q)-2pq=0

Where m is an integer, p is prime and q is prime.
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Old Mar11-09, 07:16 PM       Last edited by Kurret; Mar11-09 at 07:18 PM.. Reason: this forums spoiler function is ****            #9
Kurret

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Re: prime numbers problem

I got it.
Hint: Use the conjugate rule.


For solution
Spoiler

(p^2+q^2)/(p+q)=(p^2-q^2)+2q^2)(p+q)=p-q+q^2/((p+q)/2)

but q^2 is only divisible by 1,q,q^2. (p+q)/2 is obv not equal to 1. if it is equal to q, p=q and if it is equal to q^2, q|p and then p=q since they are prime.
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Old Mar13-09, 04:26 PM                  #10
de_brook

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Re: prime numbers problem

Originally Posted by Carl140 View Post
Hello,

I can't get this small contest problem. How do you solve this kind of problem?

Let p and q be prime numbers such that (p^2+q^2)/(p+q) is an integer.
Prove p=q.
This statement can be more generalised as follows;

Let p and q be prime numbers then (p^2+q^2)/(p+q) is a prime if and only if p = q
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Old Mar13-09, 04:31 PM                  #11
de_brook

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Re: prime numbers problem

Originally Posted by de_brook View Post
This statement can be more generalised as follows;

Let p and q be prime numbers then (p^2+q^2)/(p+q) is a prime if and only if p = q
the prime is p = q
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