Problem
NT-B1-M12-P009 Set 2. No Nonzero Solutions
#9
★★☆☆☆ Level 2 of 5
Prove that \(x^2=3y^2\) has no positive integer solutions.
If \(3\mid x^2\), then \(3\mid x\).
From \(x^2=3y^2\), we get \(3\mid x^2\), hence \(3\mid x\). Let \(x=3t\). Then \(9t^2=3y^2\), so \(y^2=3t^2\), hence \(3\mid y\). This gives a smaller solution \((t,\frac{y}{3})\). Repeating gives infinite divisibility by \(3\), impossible for positive integers.
First descent through prime divisibility.