There's a shorter proof which requires unique factorization of integers, while ignoring the assumption that a and b have no common factors.

Given a^2 = 2b^2, neither have the same number of 2s as a factor, therefore they can't be equal.

## Irrationality of sqrt(2)

There's a shorter proof which requires unique factorization of integers, while ignoring the assumption that a and b have no common factors.

Given a^2 = 2b^2, neither have the same number of 2s as a factor, therefore they can't be equal.