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January 8th, 2017, 05:01 PM  #1 
Newbie Joined: Jan 2017 From: Mangalore, Karnataka, India Posts: 11 Thanks: 0  Integer solutions to (x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)?
Whether there is any Integer solution to (x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)?

January 8th, 2017, 06:24 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,597 Thanks: 546 Math Focus: Yet to find out. 
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January 8th, 2017, 06:52 PM  #3  
Member Joined: Dec 2016 From: USA Posts: 46 Thanks: 11  Quote:
$$(x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)$$ has infinitely many integer solution triples (x,y,z). Here's one class of solutions: \begin{align*} x &= n\\ y &= \frac{n(n+1)}{2}\\ z &= \frac{n^2+n+2}{2}\\ \end{align*} where n is an arbitrary integer.  
January 8th, 2017, 07:18 PM  #4 
Newbie Joined: Jan 2017 From: Mangalore, Karnataka, India Posts: 11 Thanks: 0 
Is there any general method to find integer solutions to (ax^2+bx+c) + (ay^2+by+c) = (az^2+bz+c) ?

January 8th, 2017, 08:10 PM  #5 
Member Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 
Need the formula in General? It is very bulky. Decisions are determined using the equation Pell. For some special case solution is always there and the formula is as follows. diophantine equations  Curves triangular numbers.  Mathematics Stack Exchange 

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