factorization

[zetafunction]

1. The problem statement, all variables and given/known data

in order to factorize 20 on the rings of Q( \sqrt 2) i must solve

2. Relevant equations

x^{2} -2y^{2}=10

3. The attempt at a solution

i do not know how to solve it, i have tried by brute force with calculator but can not get any response , the given hint is that x^{2} -2y^{2}=5 has no solution.

[matt grime]

I can only guess that you've factored 20 as 2*10, so why have you not factored 10 as 2*5?

[zetafunction]

according to the teacher, who put the exercise the diophantine equation

x^{2}-2y^{2}=5 has no solution on integers.