Use Fermat's algorithm to find factors for the following integers: 175,557

I know that Fermat's algorithm shows you that, given a number n, for integers x and y, n= x^{2}+y^{2 }so n= (x+y)(x-y)

Using a method I was shown:

I took [√175,557] which is 419.

419^{2}is not 175,557,

so I took n+1 (which is 420) and computed 420^{2 }- 419^{2}= 823 which is also not a perfect square, and therefore not a factor of 175,577.

I continued this pattern for a ton of numbers, increasing by one each time, the way we had been shown in class. After 40 test numbers, I realized I am obviously doing something wrong. I would greatly appreciate it if someone could help redirect me in the correct way to use Fermat's algorithm on prime factorization.