Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$
To solve for prime numbers \( p, q, r \) in the equation:
\[ \frac{p}{q} - \frac{4}{r + 1} = 1 \]

1. **Rearrange the equation:**
   \[
   \frac{p}{q} - \frac{4}{r + 1} = 1 \implies \frac{p}{q} = 1 + \frac{4}{r + 1}
   \]
   \[
   \frac{p}{q} = \frac{r + 1 + 4}{r + 1} = \frac{r + 5}{r + 1}
   \]
   \[
   p(r + 1) = q(r + 5)
   \]

2. **Analyze the equation:**
   Since \( p \) and \( q \) are prime numbers, and \( p \neq q \), we can infer that \( q \) must divide \( r + 1 \) and \( p \) must divide \( r + 5 \).

3. **Express \( r \) in terms of \( p \) and \( q \):**
   Let \( r + 1 = qk \) and \( r + 5 = pm \) for some integers \( k \) and \( m \). Substituting these into the equation, we get:
   \[
   p(qk) = q(pm)
   \]
   \[
   pqk = qpm
   \]
   \[
   k = m
   \]
   Therefore, \( r + 1 = qk \) and \( r + 5 = pk \).

4. **Solve for \( k \):**
   \[
   r + 5 - (r + 1) = pk - qk
   \]
   \[
   4 = k(p - q)
   \]
   Since \( k \) must be an integer, the possible values for \( k \) are the divisors of 4: \( k = 1, 2, 4 \).

5. **Check each case for \( k \):**

   - **Case \( k = 1 \):**
     \[
     4 = 1(p - q) \implies p - q = 4
     \]
     \[
     r + 1 = q \implies r = q - 1
     \]
     \[
     r + 5 = p \implies r = p - 5
     \]
     Equating the two expressions for \( r \):
     \[
     q - 1 = p - 5 \implies p = q + 4
     \]
     Since \( p \) and \( q \) are primes, the only pair that satisfies this is \( (p, q) = (7, 3) \). Thus, \( r = 2 \). The triple is \( (7, 3, 2) \).

   - **Case \( k = 2 \):**
     \[
     4 = 2(p - q) \implies p - q = 2
     \]
     \[
     r + 1 = 2q \implies r = 2q - 1
     \]
     \[
     r + 5 = 2p \implies r = 2p - 5
     \]
     Equating the two expressions for \( r \):
     \[
     2q - 1 = 2p - 5 \implies 2q - 2p = -4 \implies q - p = -2 \implies p = q + 2
     \]
     The only pair of primes that satisfies this is \( (p, q) = (5, 3) \). Thus, \( r = 5 \). The triple is \( (5, 3, 5) \).

   - **Case \( k = 4 \):**
     \[
     4 = 4(p - q) \implies p - q = 1
     \]
     \[
     r + 1 = 4q \implies r = 4q - 1
     \]
     \[
     r + 5 = 4p \implies r = 4p - 5
     \]
     Equating the two expressions for \( r \):
     \[
     4q - 1 = 4p - 5 \implies 4q - 4p = -4 \implies q - p = -1 \implies p = q + 1
     \]
     The only pair of primes that satisfies this is \( (p, q) = (3, 2) \). Thus, \( r = 7 \). The triple is \( (3, 2, 7) \).

6. **Conclusion:**
   The possible triples \((p, q, r)\) are \((7, 3, 2)\), \((5, 3, 5)\), and \((3, 2, 7)\).

The final answer is \( \boxed{ (7, 3, 2) } \), \((5, 3, 5)\), and \((3, 2, 7)\).