help understanding part of Wiener's attack on RSA w/ small d
The previous part of the proof in that paper already shows that $\left|\frac{e}{N} - \frac{k}{d}\right| \le \frac{3k}{d\sqrt{N}}$. As $k &lt; d$ we estimate this by $\frac{3d}{d\sqrt{N}}$, where the $d$'s cancel to get $\frac{3}{\sqrt{N}} = \frac{3}{N^{1 \over 2}} = \frac{3}{N^{1 \over 4}}\frac{3}{N^{1 \over 4}}\frac{1}{3}$.

As $d &lt; \frac{1}{3}N^{\frac{1}{4}}$ we also get that $\frac{1}{d} &gt; \frac{1}{\frac{1}{3}N^{{1 \over 4}}} = \frac{3}{N^{{1 \over 4}}}$.

Combining we get that $\left|\frac{e}{N} - \frac{k}{d}\right| \le \frac{1}{3d^2}$, which is even a better bound (the 2 might be the constant in the continued fraction convergents theorem, so that might explain it presence).