Problem
NT-B1-M06-P011 The Equation \(x^2+y^2=3z^2\)
#11
★★☆☆☆ Level 2 of 5
Prove that \(x^2+y^2=3z^2\) has no positive integer solutions.
Use the previous problem, then divide all variables by \(3\).
From \(3\mid x^2+y^2\), we get \(3\mid x\) and \(3\mid y\). Let \(x=3x_1\), \(y=3y_1\). Then \(9x_1^2+9y_1^2=3z^2\), hence \(z^2=3x_1^2+3y_1^2\), so \(3\mid z\). Dividing \(x,y,z\) by \(3\) gives a smaller positive solution. Infinite descent is impossible.
A classical model of modular descent.