Find all solutions to the Diophantine equation x^2+y^2 = 7z^2.
Please teach me how to solve it . Thank you.
7 is a number of the form 4k+3
can have only two form, .
has the form of 4k+3.
The important thing here is that the sum of two squares,
can never take the form 4k+3.
Thus, there are no solutions to this equation.
I got it . Thank you very much.
Also, how should I prove the below question?
Prove Fermat's last theorem for exponent three; i.e. prove that if
x^3 + y^3 = z^3
where x, y, and z are rational integers, then x, y, or z is 0.
Hint: show that x^3 + y^3 = (epsilon)z^3 , where x, y, and z are quadratic integers in Q[-3], and epsilon is a unit in Q[-3] , then x, y, or z is 0.