The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive of it. Prove that the resulting sum is the squares of an integer.

Explanation

To prove that the sum of the fourth power of the common difference of an arithmetic progression (AP) and the product of any four consecutive terms of that AP is the square of an integer, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Arithmetic Progression:**
Let the first term of the AP be \( A \) and the common difference be \( D \). The four consecutive terms can be represented as:
\[
A - 3D, \quad A - D, \quad A + D, \quad A + 3D
...