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January 8th, 2017, 06:01 PM   #1
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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)?
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January 8th, 2017, 07:24 PM   #2
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January 8th, 2017, 07:52 PM   #3
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Quote:
Originally Posted by Naveenchandra Kumar View Post

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.
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January 8th, 2017, 08:18 PM   #4
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Is there any general method to find integer solutions to (ax^2+bx+c) + (ay^2+by+c) = (az^2+bz+c) ?
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January 8th, 2017, 09:10 PM   #5
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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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