SOLUTION: Find all positive integers $n$ for which $n^2

SOLUTION: Find all positive integers $n$ for which $n^2 - 19n + 99$ is a perfect square.

Question 1074625: Find all positive integers $n$ for which $n^2 - 19n + 99$ is a perfect square.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
TIP:
When I see $ wrapping equations,
it remind me of LATEX,
and makes me suspect you come from artofproblemsolving.
We use triple curly brackets, like {, here.
I am going to look at the embarrassingly simpler solution in that website
as soon as I finish typing my awkward solution here.

ONE WAY:
We want to be true
for some positive integers and .

Let me abbreviate with the change of variables
,
knowing that is a positive, even integer,
and is an integer.
The equation becomes
, so much easier to write,
and immediately we think of
.
Are those factors in brackets positive or negative?
We knew that ,
so --> --> ,
so whether or ,
and .
Back to the product of positive integer factors
.
The factors could be 1 and 35 or 5 and 7.
We have 4 possibilities:
-->-->-->
-->-->-->
-->-->-->
-->-->-->

ANOTHER WAY:
<--> , so if ,
-->--> .
The two perfect squares we look for, and ,
cannot be too large if their difference is only .
There are not many possible solutions,
and they will all be close to the of the vertex.
So, listing values for around ,
(or listing values for for around ),
until you find a perfect square is an option.

PS: Actually, it was not that the artofproblemsolving solution was so much simpler,
but that they skip steps/explanations that "should be obvious,"
so they do not need to write as much.
If you get all those steps, artofproblemsolving is a fun site,
but I tend to/like to overexplain, so I belong at this website.
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