Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying
\[m(m + k) = n(n + 1).\]
To find all positive integers \( k \) such that there exist two positive integers \( m \) and \( n \) satisfying
\[ m(m + k) = n(n + 1), \]
we will analyze different cases for \( k \).

1. **Case \( k = 1 \):**
   \[ m(m + 1) = n(n + 1). \]
   If we take \( m = 1 \) and \( n = 1 \), we get:
   \[ 1(1 + 1) = 1(1 + 1) \implies 2 = 2. \]
   Thus, \( k = 1 \) is a solution.

2. **Case \( k = 2 \):**
   \[ m(m + 2) = n(n + 1). \]
   We can rewrite this equation as:
   \[ m^2 + 2m = n^2 + n. \]
   Rearranging terms, we get:
   \[ m^2 + 2m - n^2 - n = 0. \]
   This can be written as:
   \[ (2m - 2n + 1)(2m + 2n + 3) = 3. \]
   Since \( 2m + 2n + 3 \geq 7 \), this equation has no solutions in positive integers. Therefore, \( k = 2 \) is not a solution.

3. **Case \( k = 3 \):**
   \[ m(m + 3) = n(n + 1). \]
   We can rewrite this equation as:
   \[ m^2 + 3m = n^2 + n. \]
   Rearranging terms, we get:
   \[ m^2 + 3m - n^2 - n = 0. \]
   This can be written as:
   \[ (2m - 2n + 2)(2m + 2n + 4) = 8. \]
   Since \( 2m + 2n + 4 \geq 8 \) and \( 2m - 2n + 2 \neq 1 \), this equation has no solutions in positive integers. Therefore, \( k = 3 \) is not a solution.

4. **Case \( k = 2p \geq 4 \):**
   We need to find \( m \) and \( n \) such that:
   \[ m(m + 2p) = n(n + 1). \]
   If we take \( m = p^2 - p \) and \( n = p^2 - 1 \), we get:
   \[ (p^2 - p)(p^2 - p + 2p) = (p^2 - 1)(p^2 - 1 + 1). \]
   Simplifying, we get:
   \[ (p^2 - p)(p^2 + p) = (p^2 - 1)p^2. \]
   \[ p^4 - p^2 = p^4 - p^2. \]
   Thus, \( k = 2p \geq 4 \) is a solution.

5. **Case \( k = 2p + 1 \geq 5 \):**
   We need to find \( m \) and \( n \) such that:
   \[ m(m + 2p + 1) = n(n + 1). \]
   If we take \( m = \frac{p(p - 1)}{2} \) and \( n = \frac{p(p + 1)}{2} - 1 \), we get:
   \[ \left(\frac{p(p - 1)}{2}\right)\left(\frac{p(p - 1)}{2} + 2p + 1\right) = \left(\frac{p(p + 1)}{2} - 1\right)\left(\frac{p(p + 1)}{2}\right). \]
   Simplifying, we get:
   \[ \left(\frac{p(p - 1)}{2}\right)\left(\frac{p(p - 1) + 4p + 2}{2}\right) = \left(\frac{p(p + 1) - 2}{2}\right)\left(\frac{p(p + 1)}{2}\right). \]
   \[ \left(\frac{p(p - 1)}{2}\right)\left(\frac{p^2 + 3p + 2}{2}\right) = \left(\frac{p^2 + p - 2}{2}\right)\left(\frac{p^2 + p}{2}\right). \]
   \[ \frac{p(p - 1)(p^2 + 3p + 2)}{4} = \frac{(p^2 + p - 2)(p^2 + p)}{4}. \]
   \[ p(p - 1)(p^2 + 3p + 2) = (p^2 + p - 2)(p^2 + p). \]
   \[ p^4 + 3p^3 + 2p^2 - p^3 - 3p^2 - 2p = p^4 + p^3 - 2p^2 + p^3 + p^2 - 2p. \]
   \[ p^4 + 2p^3 - p^2 - 2p = p^4 + 2p^3 - p^2 - 2p. \]
   Thus, \( k = 2p + 1 \geq 5 \) is a solution.

Hence, the answer is:
\[ \boxed{\mathbb{N} \setminus \{2, 3\}}. \]