(c) $p=4 m+1$ for some $m \in \mathbb{Z}$ if and only if $p$ is an odd prime and $p$ can be written as a sum of integers squared.

Step 1 ：Define a function to check if a number is a prime number.

Step 2 ：Define a function to check if a number is odd.

Step 3 ：Define a function to check if a number can be written as a sum of two squares.

Step 4 ：Define a function to check if a number is of the form \(4m+1\).

Step 5 ：Define a function to check all these conditions for a number.

Step 6 ：Check the conditions for the numbers 5, 13, 17, and 19.

Step 7 ：The output is True for the numbers 5, 13, and 17, which means these numbers satisfy all the conditions. They are prime numbers, they are odd, they can be written as a sum of two squares, and they are of the form \(4m+1\).

Step 8 ：The output is False for the number 19, which means this number does not satisfy all the conditions.

Step 9 ：\(\boxed{\text{The numbers 5, 13, and 17 satisfy all these conditions, while the number 19 does not.}}\)