Which of these integers can be written as the sum of two squares of integers?

Question

I know that an integer is the sum of two squares if and only if it is the norm of some Gaussian integer. However, the numbers below are so large I don't even know where to begin, let alone know whether or not it can even be a sum of two squares.

1. 5^7 * 7^9 * 11^12 * 13^11
2. 2^10 * 11^12 * 13^15 * 23^11
3. 3^11 * 7^10 * 13^12 * 17^13
4. 3^8 * 7^12 * 13^5 * 19^4

Answer 1

By Fermat's theorem, an integer is the sum of two square numbers iff each of its prime factors of the form $4k+3$ has an even power in its prime factorization.

This should lead you to conclude.