Pairs of integers

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SANZEED
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Pairs of integers

Unread post by SANZEED » Sat Mar 23, 2013 11:45 pm

Find infinitely many pairs of integers $(a,b)$ such that $1<a<b$ and $ab$ divides $a^{2}+b^{2}-1$. Also find all positive integers $k$ such that there exists $(a,b)$ such that $\frac {a^{2}+b^{2}-1}{ab}=k$.
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Phlembac Adib Hasan
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Re: Pairs of integers

Unread post by Phlembac Adib Hasan » Sun Mar 24, 2013 8:58 am

SANZEED wrote:Find infinitely many pairs of integers $(a,b)$ such that $1<a<b$ and $ab$ divides $a^{2}+b^{2}-1$. Also find all positive integers $k$ such that there exists $(a,b)$ such that $\frac {a^{2}+b^{2}-1}{ab}=k$.
Take $a>1,\; b=a^2-1$. Then $\dfrac {a^2+b^2-1}{ab}=a=$ Any integer greater than $1$. $\dfrac {a^2+b^2-1}{ab}=1$ is not possible because it will imply $(a-b)^2=1-ab<0$ since $ab>1$. So all the $\{k\}=\mathbb N\backslash 1$
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harrypham
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Re: Pairs of integers

Unread post by harrypham » Sat Aug 10, 2013 7:30 am

SANZEED wrote:Find infinitely many pairs of integers $(a,b)$ such that $1<a<b$ and $ab$ divides $a^{2}+b^{2}-1$. Also find all positive integers $k$ such that there exists $(a,b)$ such that $\frac {a^{2}+b^{2}-1}{ab}=k$.
We use Vieta Jumping to solve this problem. ;)

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