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Find all solutions in integers $m,n$ for the equation $(m-n)²=4mn/(m+n-1)$ I tried solving this but got stuck at one point , after simplification this equation reduces to, $(m-n)²=m+n$, after this how do we conclude that $m=k(k+1)/2$,$n=k(k-1)/2$, $k$ is integer greater or equal to 2?

rschwieb
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Dhruva
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  • Hi Dhruva. We have a Latex-like typesetting system for mathematical expressions, called MathJax. Information is here: https://math.meta.stackexchange.com/a/10164 – 311411 Feb 12 '22 at 02:38

1 Answers1

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Denote $u=m+n, k=m-n$. Then you have got $k^2=u$. So $m=(u+k)/2=(k^2+k)/2, n=(u-k)/2=(k^2-k)/2$, Q.E.D.

markvs
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