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November 15th, 2017, 01:55 PM   #1
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Diophantine hyperbola

I am given an integer c an and we know the sum of consecutives integers can reach this number. I reach this expression
\begin{align*}
c &= \dfrac{n(n+1)}{2} - \dfrac{m(m+1)}{2}
\end{align*}

where n > m, both are integers positive.

we can reduce that to
\begin{align*}
2 * c &= n^2 + n - m^2 - m
\end{align*}

For example if c = 1000 then one of the solutions is n = 52 and m = 27.
How can I get all the integer solutions n and m ?

Last edited by limboa; November 15th, 2017 at 02:04 PM. Reason: v8archie was right
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November 15th, 2017, 02:01 PM   #2
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${}-m$, not ${}+m$
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November 15th, 2017, 02:55 PM   #3
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Not sure it helps, but \begin{align*}n^2+n-m^2-m&=\frac14\left((2n+1)^2 - (2m+1)^2\right) \\ &=\frac14(2n+2m+2)(2n-2m)=(n+m+1)(n-m) \end{align*}
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