Can be a complete square for some integer x > 0?
Sorry I should add a condition x > 0 in the subject line.
We want to solve:
Complete the square by multiplying both sides by and adding and subtracting :
Let :
Factor:
Now let's factor :
So you can see, we only have a finite number of cases to consider ( to be exact) when solving for and
e.g. is one example.
I'll leave you to consider all these cases, but you should find that , or in other words , is the only time there is a solution in .
Thus for , is never square.