A note on class number one criteria of Sirola for real quadratic fields

In a recent paper, Sirola gives two necessary and sufficient conditions for the class number of a real quadratic field to be equal to one. The purpose of this note is to remark that the equivalence of these conditions can be proved by using an elementary result of Nagell, which itself is a simple consequence of the fact that the Pell equation X2 - dY2 = 1 always has solutions in positive integers when d > 1 is squarefree