Given:
The equation is m^2 - 105 = n^2.

To find:
The number of pairs (m, n) of positive integers that satisfy the equation.

Solution:
To solve this problem, let's analyze the equation step by step.

Step 1:
Rearranging the equation, we get:
m^2 - n^2 = 105

Step 2:
We can further simplify the equation using the difference of squares formula:
(m + n)(m - n) = 105

Step 3:
Now, we need to find all the factors of 105 and pair them up in such a way that their difference is an even number.

The factors of 105 are:
1, 3, 5, 7, 15, 21, 35, 105

Step 4:
Let's pair up the factors:
(1, 105), (3, 35), (5, 21), (7, 15)

Step 5:
Now, we need to find the values of m and n for each pair of factors.

For the first pair (1, 105), we have:
m + n = 105
m - n = 1

Adding the two equations, we get:
2m = 106
m = 53

Substituting the value of m in one of the equations, we get:
53 + n = 105
n = 52

Therefore, the pair (m, n) is (53, 52).

Step 6:
Similarly, we can find the values of m and n for the other pairs of factors:

For the pair (3, 35), we have:
m + n = 35
m - n = 3

Adding the two equations, we get:
2m = 38
m = 19

Substituting the value of m in one of the equations, we get:
19 + n = 35
n = 16

Therefore, the pair (m, n) is (19, 16).

Step 7:
For the pair (5, 21), we have:
m + n = 21
m - n = 5

Adding the two equations, we get:
2m = 26
m = 13

Substituting the value of m in one of the equations, we get:
13 + n = 21
n = 8

Therefore, the pair (m, n) is (13, 8).

Step 8:
For the pair (7, 15), we have:
m + n = 15
m - n = 7

Adding the two equations, we get:
2m = 22
m = 11

Substituting the value of m in one of the equations, we get:
11 + n = 15
n = 4

Therefore, the pair (m, n) is (11, 4).

Step 9:
Thus, there are a total of 4 pairs (m, n) of positive integers that satisfy the equation m^2 - 105 = n^2. These pairs are: