Why can't 57 be written as the sum of two squares?

I was under the impression that only numbers of the form 3 mod 4 could not be written in this way, and 57 does not equal 3 mod 4? Unless I'm missing something majorly obvious

Thanks

Question

Why can't 57 be written as the sum of two squares?

I was under the impression that only numbers of the form 3 mod 4 could not be written in this way, and 57 does not equal 3 mod 4? Unless I'm missing something majorly obvious

Thanks

kc_kennylau · Answer

Because
1+1=2, 1+4=5, 1+9=10, 1+16=17, 1+25=26, 1+36=37, 1+49=50,
4+4=8, 4+9=13, 4+16=20, 4+25=29, 4+36=40, 4+49=53,
9+9=18, 9+16=25, 9+25=34, 9+36=44,
16+16=32, 16+25=31, 16+36=52,
25+25=50

kc_kennylau · Answer

I don't really think that you can predict which numbers can be written as the sum of two squares...

kc_kennylau · Answer

Wait, something from Wikipedia: A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k + 3. This is generalized by Waring's problem.

kc_kennylau · Answer

19 is in the form of 4k + 3

anonymous · Answer

ahh okay, it's the prime factorisation which can't be of the form 3 mod 4. Great thanks for that!!