# Thread: square of an integer

1. ## square of an integer

Show that $2004^2+2004^2\cdot2005^2+2005^2$ is the square of an integer.

Any hint will be appreciated.

2. Hello,
Originally Posted by disclaimer
Show that $2004^2+2004^2\cdot2005^2+2005^2$ is the square of an integer.

Any hint will be appreciated.
It can be proved in general :
$N^2+N^2 \cdot (N+1)^2+(N+1)^2$ is a square.

All you need to know is :
$(a-b)^2=a^2+b^2-2ab$
$(a+b)^2=a^2+b^2+2ab$

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$N^2+(N+1)^2=\underbrace{N^2+(N+1)^2-2N(N+1)}_{[(N+1)-N]^2}+2N(N+1)$

So $N^2+N^2 \cdot (N+1)^2+(N+1)^2=[(N+1)-N]^2+2N(N+1)+[N(N+1)]^2$

$=1+2N(N+1)+[N(N+1)]^2=\boxed{[N(N+1)+1]^2}$

3. Thank you, Moo.