Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers, 
then it can't take value 4 for any integer.
1. **Claim**: The lowest positive integer \( n \) such that if a polynomial with integer coefficients takes the value 2 for \( n \) different integers, then it cannot take the value 4 for any integer, is \( n = 4 \).

2. **Example for \( n = 3 \)**: Consider the polynomial \( P(x) = (x-1)(x-3)(x-4) + 2 \). 
   - For \( x = 1 \), \( P(1) = (1-1)(1-3)(1-4) + 2 = 0 + 2 = 2 \).
   - For \( x = 3 \), \( P(3) = (3-1)(3-3)(3-4) + 2 = 0 + 2 = 2 \).
   - For \( x = 4 \), \( P(4) = (4-1)(4-3)(4-4) + 2 = 0 + 2 = 2 \).
   - However, for \( x = 2 \), \( P(2) = (2-1)(2-3)(2-4) + 2 = (-1)(-2) + 2 = 2 + 2 = 4 \).

   This shows that for \( n = 3 \), the polynomial can take the value 4 for some integer \( x \). Therefore, \( n > 3 \).

3. **General Case for \( n \geq 4 \)**: Suppose \( n \geq 4 \) and the property is false. Let \( P(x) \) be a polynomial for which the property fails. Then, we can write:
   \[
   P(x) = \alpha (x - r_1)(x - r_2) \cdots (x - r_n) + 2
   \]
   where \( r_1, r_2, \ldots, r_n \) are distinct integers. Since the property fails, there exists some integer \( m \) such that \( P(m) = 4 \). Therefore:
   \[
   4 = \alpha (m - r_1)(m - r_2) \cdots (m - r_n) + 2
   \]
   which simplifies to:
   \[
   2 = \alpha (m - r_1)(m - r_2) \cdots (m - r_n)
   \]

4. **Analysis of \( \alpha (m - r_1)(m - r_2) \cdots (m - r_n) = 2 \)**: Since \( 2 = \alpha (m - r_1)(m - r_2) \cdots (m - r_n) \), the product \( (m - r_1)(m - r_2) \cdots (m - r_n) \) must be either \( 2 \) or \( -2 \). This implies that each \( |m - r_i| \) must be either 1 or 2.

5. **Contradiction**: Since \( n \geq 4 \), there are at least 4 distinct \( r_i \)'s. At most one of the \( |m - r_i| \) can be 2, so at least three of the \( |m - r_i| \) must be 1. This means there exist \( i \neq j \) such that \( m - r_i = m - r_j \), which contradicts the distinctness of \( r_i \) and \( r_j \).

6. **Conclusion**: Therefore, the property never fails for \( n = 4 \), as desired.

\(\blacksquare\)

The final answer is \( \boxed{ n = 4 } \).