Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.
To prove that \( a > n^4 - n^3 \), we start by analyzing the given conditions and using the properties of divisibility.

1. **Given Conditions**:
   - \( n > 1 \) and \( a > n^2 \).
   - Among the integers \( a+1, a+2, \ldots, a+n \), there is a multiple of each of the numbers \( n^2+1, n^2+2, \ldots, n^2+n \).

2. **Divisibility Condition**:
   Let \( n^2 + i \mid a + x_i \) for \( i = 1, 2, \ldots, n \). This means that \( a + x_i \equiv 0 \pmod{n^2 + i} \).

3. **Constructing the Expression**:
   Consider two consecutive terms \( n^2 + i \) and \( n^2 + i + 1 \). We have:
   \[
   n^2 + i \mid a + x_i \quad \text{and} \quad n^2 + i + 1 \mid a + x_{i+1}
   \]
   We can write:
   \[
   (n^2 + i)(n^2 + i + 1) \mid (a + x_i)(n^2 + i + 1) - (a + x_{i+1})(n^2 + i)
   \]
   Simplifying the right-hand side:
   \[
   (a + x_i)(n^2 + i + 1) - (a + x_{i+1})(n^2 + i) = a(n^2 + i + 1) + x_i(n^2 + i + 1) - a(n^2 + i) - x_{i+1}(n^2 + i)
   \]
   \[
   = a + x_i(n^2 + i + 1) - x_{i+1}(n^2 + i)
   \]
   \[
   = a + x_i(n^2 + i + 1) - x_{i+1}(n^2 + i)
   \]

4. **Analyzing the Expression**:
   We need to ensure that the expression is divisible by \( (n^2 + i)(n^2 + i + 1) \). This implies:
   \[
   (n^2 + i)^2 + (n^2 + i) \leq |a + x_i + (n^2 + i)(x_i - x_{i+1})|
   \]

5. **Bounding \( a \)**:
   If the quantity inside the modulus is positive for some \( i \), then:
   \[
   a \geq (n^2 + i)^2 + (n^2 + i)(1 + x_{i+1} - x_i) - x_i
   \]
   To get the maximum negative value, we set \( x_i = n \) and \( x_{i+1} = 1 \):
   \[
   a \geq (n^2 + i)^2 + (n^2 + i)(1 - n + 1) + n
   \]
   Simplifying:
   \[
   a \geq (n^2 + i)^2 + (n^2 + i)(2 - n) + n
   \]
   \[
   a \geq n^4 - n^3 + 2n^2(i + 1) + 2i - ni
   \]

6. **Conclusion**:
   Since \( 2n^2(i + 1) + 2i - ni \) is positive, we conclude that:
   \[
   a > n^4 - n^3
   \]

The final answer is \( \boxed{ a > n^4 - n^3 } \)