My Math Forum Integer solutions to (x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)?

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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: 1,361 Thanks: 467 Math Focus: Yet to find out. What happens if x=y Sent from my iPhone using Tapatalk
January 8th, 2017, 06:52 PM   #3
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Quote:
 Originally Posted by Naveenchandra Kumar Whether there is any Integer solution to (x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)?
The equation
$$(x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2)$$
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 Newbie   Joined: Jul 2015 From: tbilisi Posts: 27 Thanks: 7 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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