A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0.

a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared".

b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 

1) $e-2$ , $e$ , $e+4$ are all prime numbers.
2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

### Part (a)
We need to prove that 2020 can be written as a sum of 2, 3, and 4 perfect squares.

1. **2-squared:**
   \[
   2020 = 16^2 + 42^2
   \]
   Calculation:
   \[
   16^2 = 256, \quad 42^2 = 1764
   \]
   \[
   256 + 1764 = 2020
   \]
   Thus, 2020 is 2-squared.

2. **3-squared:**
   \[
   2020 = 18^2 + 20^2 + 36^2
   \]
   Calculation:
   \[
   18^2 = 324, \quad 20^2 = 400, \quad 36^2 = 1296
   \]
   \[
   324 + 400 + 1296 = 2020
   \]
   Thus, 2020 is 3-squared.

3. **4-squared:**
   \[
   2020 = 12^2 + 16^2 + 18^2 + 36^2
   \]
   Calculation:
   \[
   12^2 = 144, \quad 16^2 = 256, \quad 18^2 = 324, \quad 36^2 = 1296
   \]
   \[
   144 + 256 + 324 + 1296 = 2020
   \]
   Thus, 2020 is 4-squared.

### Part (b)
We need to determine all natural numbers \(a, b, c, d, e\) such that \(a < b < c < d < e\) and the following conditions hold:
1. \(e-2\), \(e\), \(e+4\) are all prime numbers.
2. \(a^2 + b^2 + c^2 + d^2 + e^2 = 2020\).

1. **Finding \(e\):**
   From condition (1), we need \(e-2\), \(e\), and \(e+4\) to be prime numbers. The possible values for \(e\) are:
   \[
   e = 43 \quad \text{(since 41, 43, and 47 are all prime)}
   \]

2. **Finding \(a, b, c, d\):**
   Given \(e = 43\), we need:
   \[
   a^2 + b^2 + c^2 + d^2 + 43^2 = 2020
   \]
   Calculation:
   \[
   43^2 = 1849
   \]
   \[
   a^2 + b^2 + c^2 + d^2 = 2020 - 1849 = 171
   \]

3. **Checking possible combinations:**
   We need to find \(a, b, c, d\) such that \(a < b < c < d\) and \(a^2 + b^2 + c^2 + d^2 = 171\). By checking possible combinations, we find:
   \[
   (a, b, c, d) = (4, 5, 7, 9), (1, 5, 8, 9), (3, 4, 5, 11)
   \]

Thus, the solutions are:
\[
\boxed{(a, b, c, d, e) = (4, 5, 7, 9, 43), (1, 5, 8, 9, 43), (3, 4, 5, 11, 43)}
\]