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 November 15th, 2017, 01:55 PM #1 Newbie   Joined: Nov 2017 From: Mexico Posts: 1 Thanks: 0 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
 November 15th, 2017, 02:01 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra ${}-m$, not ${}+m$
 November 15th, 2017, 02:55 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 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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