Find the ordered pairs

• Sep 18th 2010, 11:45 AM
spoc21
Find the ordered pairs
Hello, I'm having some trouble with the following question, and have not been able to solve it despite excessive attempts:

Find all ordered pairs of integers (x,y) such that \$\displaystyle x^2+2x+18=y^2 \$

I tried square-rooting both sides so that the equation could be solved for y, but didn't have any luck. I also thought about using the "completing the square" method, but am not sure on how to use it in this case. I would greatly appreciate any help.

Thanks!
• Sep 18th 2010, 12:00 PM
Plato
I found one pair. There may be others.
Look at \$\displaystyle (x+1)^2=y^2-17\$.
• Sep 18th 2010, 01:26 PM
HallsofIvy
Plato's suggestion, that you rewrite the equation as \$\displaystyle x^2+ 2x+ 1- 1+ 18= y^2\$ so that \$\displaystyle (x+ 1)^2= y^2- 17 \$ is excellent! If you let z= x+1, then that equation becomes \$\displaystyle z^2= y^2- 17\$ or \$\displaystyle y^2- z^2= (y- z)(y+ z)= 17\$. Since y and z are integers, you should be able to quickly find Plato's solution as well as the three other ordered pairs satifying that equation.