A number theory problem by Rahil Sehgal

$n^2 + 20n +15 = m^2$

$(n + 10)^2 - 85 = m^2$

$\text {Let } k = n + 10 \Rightarrow n = k - 10$

$k^2 - m^2 = 85$

Now, we can see that |k| > |m|

$(k + m)(k - m) = 85$

The factors of 85 are: ±1, ±85, ±5 and ±17.

$\text {We can form the following } (f_1, f_2) = (k + m, k - m), f_1 × f_2 = 85 \text { pairs: }$

(85, 1) ; (-85, -1) ; (17, 5) and (-17, -5) (and vice versa, however the reversed pairs won't give us new solutions for k (and n), we would only need those, if we had to determine the m values as well).

$f_1 + f_2 = (k + m) + (k - m) = 2k \Rightarrow k = \frac {f_1 + f_2 }{2}$

This gives us the following 4 solutions for k: ± 43 and ± 11 .

It is easy to see now, that the sum of the distinct k values is 0.

Hence, thee sum of the distinct n values is:

$0 - 4×10 = \boxed {-40}$