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April 15th, 2017, 10:16 AM  #1 
Senior Member Joined: Sep 2015 From: USA Posts: 1,656 Thanks: 842  one of these that JohnG is so good at
show that $\forall a,b \in \mathbb{N}$ $m = \dfrac{a^2 + b^2}{ab+1} \in \mathbb{N} \Rightarrow \sqrt{m}\in \mathbb{N}$ 
April 16th, 2017, 03:16 AM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,460 Thanks: 489 Math Focus: Yet to find out. 
JohnG? I think numberphile showcased this problem.

April 16th, 2017, 05:09 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,250 Thanks: 1439 
Let $n = \sqrt{m}$. If $n$ > 1, must $n^3$ divide $a$ or $b$, but not both?

April 16th, 2017, 10:52 AM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 1,656 Thanks: 842  Quote:
Last edited by romsek; April 16th, 2017 at 10:55 AM.  
April 17th, 2017, 09:40 AM  #7 
Member Joined: Jan 2016 From: Athens, OH Posts: 62 Thanks: 37 
JohnG is good at easy number theory problems. I'm glad I don't have to think about this one.


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good, johng 
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