Some positive integers are initially written on a board, where each $2$ of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$
(2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$
After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that:
Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

1. **Initial Setup and Moves**:
   - We start with some positive integers on the board, each pair of them being different.
   - We can perform two types of moves:
     1. If there are numbers \( n \) and \( n+1 \) on the board, we can replace them with \( n-2 \).
     2. If there are numbers \( n \) and \( n+4 \) on the board, we can replace them with \( n-1 \).

2. **Example for \( c = -3 \)**:
   - Consider the sequence \( 1, 2, 3, 4, 5 \):
     \[
     \begin{align*}
     1, 2, 3, 4, 5 &\rightarrow 0, 2, 3, 4 \quad (\text{using } 1 \text{ and } 2) \\
     0, 2, 3, 4 &\rightarrow 0, 1, 2 \quad (\text{using } 3 \text{ and } 4) \\
     0, 1, 2 &\rightarrow -2, 2 \quad (\text{using } 0 \text{ and } 1) \\
     -2, 2 &\rightarrow -3 \quad (\text{using } -2 \text{ and } 2)
     \end{align*}
     \]
   - This shows that we can reach \( -3 \) using the given moves.

3. **Defining Powers**:
   - Define the "A-power" of an integer \( n \) as \( a_n \), where \( a_0 = 1 \) and \( a_n + a_{n+1} = a_{n-2} \).
   - Define the "B-power" of an integer \( n \) as \( b_n \), where \( b_0 = 1 \) and \( b_n + b_{n+4} = b_{n-1} \).
   - The A-power and B-power of a board \( C \) are the sums of the A-powers and B-powers of the numbers on it, respectively.

4. **Characteristic Polynomials**:
   - The characteristic polynomial for \( a_n \) is \( f_a(x) = x^3 + x^2 - 1 \).
   - The characteristic polynomial for \( b_n \) is \( f_b(x) = x^5 + x - 1 \).
   - Both polynomials have the same real root \( r \), since \( (x^3 + x^2 - 1)(x^2 - x + 1) = x^5 + x - 1 \).

5. **Invariance of Power**:
   - The power of the board \( P(C) = \sum_{i \in (\text{numbers on board})} r^i \) is invariant under the given moves.
   - For any board \( C \), \( P(C) < \sum_{i=1}^{\infty} r^i \):
     \[
     P(C) < \frac{r}{1-r}
     \]

6. **Contradiction for \( n \leq -4 \)**:
   - Suppose there is a number \( n \leq -4 \) on the board. This implies \( P(C) \geq r^{-4} \).
   - Using the equation \( r^5 + r - 1 = 0 \):
     \[
     r^5 = 1 - r \implies 1 = \frac{r^5}{1-r} \implies r^{-4} = \frac{r}{1-r}
     \]
   - Thus, \( P(C) \geq \frac{r}{1-r} \), which contradicts \( P(C) < \frac{r}{1-r} \).

7. **Conclusion**:
   - Therefore, there can be no integer on the board with value less than or equal to \( -4 \).
   - The maximum value of \( c \) such that each number appearing in any move is greater or equal to \( c \) is \( -3 \).

The final answer is \( \boxed{ c = -3 } \)