examday

/TMUA

The following lines can be arranged to prove that there are no positive integers $a$ and $b$ satisfying

• P. Suppose that such pairs exist, and choose one for which $a$ is as small as possible.
• Q. Since $a^2$ is even, $a$ is even; write $a=2c$.
• R. Substitution gives $b^2=2c^2$.
• S. Hence $b$ is even; write $b=2d$.
• T. Substitution now gives $c^2=2d^2$.
• U. But $c=a/2<a$, so $(c,d)$ is a smaller positive-integer solution. This is a contradiction.

Which is the correct order?