Problem
NT-B1-M06-P019 Minimal Denominator
#19
★★★☆☆ Level 3 of 5
Prove that there do not exist positive integers \(a,b\) with minimal possible \(b\) such that \(\left(\frac{a}{b}\right)^2=2\).
From \(a^2=2b^2\), obtain a new representation with a smaller denominator.
From \(a^2=2b^2\), both \(a\) and \(b\) are even. Then \(a=2a_1\), \(b=2b_1\), and \(\left(\frac{a_1}{b_1}\right)^2=2\). But \(b_1=\frac{b}{2}
This is the same irrationality proof in explicit minimal-counterexample form.