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 November 15th, 2017, 12:55 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 01:04 PM. Reason: v8archie was right November 15th, 2017, 01:01 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra ${}-m$, not ${}+m$ November 15th, 2017, 01:55 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 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*} Tags diophantine, hyperbola Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post hyperbola Pre-Calculus 6 December 18th, 2014 06:56 PM johnny Algebra 2 February 27th, 2011 03:53 PM mad Algebra 7 August 20th, 2010 01:50 AM mikeportnoy Algebra 1 December 6th, 2009 04:37 AM gretchen Algebra 2 February 15th, 2007 03:37 PM

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