Find the sum of all those integers n for which `n^2+20n+15` is the square of an integer.

Answer

To solve the problem of finding the sum of all integers \( n \) for which \( n^2 + 20n + 15 \) is the square of an integer, we can follow these steps:

### Step 1: Set up the equation
Let \( S(n) = n^2 + 20n + 15 \). We want to find integers \( n \) such that \( S(n) = m^2 \) for some integer \( m \).

### Step 2: Rewrite the expression
We can rewrite \( S(n) \) as:
\[ S(n) = (n + 10)^2 - 100 + 15 = (n + 10)^2 - 85 \]
Thus, we have:
\[ m^2 = (n + 10)^2 - 85 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ (n + 10)^2 - m^2 = 85 \]
This can be factored using the difference of squares:
\[ (n + 10 - m)(n + 10 + m) = 85 \]

### Step 4: Factor 85
Next, we find the pairs of factors of 85. The factor pairs of 85 are:
1. \( (1, 85) \)
2. \( (5, 17) \)
3. \( (-1, -85) \)
4. \( (-5, -17) \)

### Step 5: Solve for each factor pair
For each factor pair \( (a, b) \), we set:
\[ n + 10 - m = a \quad \text{and} \quad n + 10 + m = b \]

#### Case 1: \( (1, 85) \)
\[ n + 10 - m = 1 \quad \text{and} \quad n + 10 + m = 85 \]
Adding these equations:
\[ 2(n + 10) = 86 \implies n + 10 = 43 \implies n = 33 \]

#### Case 2: \( (5, 17) \)
\[ n + 10 - m = 5 \quad \text{and} \quad n + 10 + m = 17 \]
Adding these equations:
\[ 2(n + 10) = 22 \implies n + 10 = 11 \implies n = 1 \]

#### Case 3: \( (-1, -85) \)
\[ n + 10 - m = -1 \quad \text{and} \quad n + 10 + m = -85 \]
Adding these equations:
\[ 2(n + 10) = -86 \implies n + 10 = -43 \implies n = -53 \]

#### Case 4: \( (-5, -17) \)
\[ n + 10 - m = -5 \quad \text{and} \quad n + 10 + m = -17 \]
Adding these equations:
\[ 2(n + 10) = -22 \implies n + 10 = -11 \implies n = -21 \]

### Step 6: Collect all values of \( n \)
The integer values of \( n \) we found are:
- \( n = 33 \)
- \( n = 1 \)
- \( n = -53 \)
- \( n = -21 \)

### Step 7: Calculate the sum of all \( n \)
Now, we calculate the sum:
\[ 33 + 1 - 53 - 21 = -40 \]

### Final Answer
The sum of all integers \( n \) for which \( n^2 + 20n + 15 \) is a perfect square is \( -40 \).

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