# Congratulations, Bill! [duplicate]

Only a short while after returning, he has broken the 100k!

• Anyway: $$\rm A\, hearty\, and\, well-earned\, congratulations\, Bill!\$$ I appreciated so much your answering style: $$\rm \ \ n=3k\!-\!2\!:\ {\rm mod}\,\ \color{#0a0}{d_1:\ a^3\equiv b^2},\, b^3\equiv 1\,\Rightarrow\, a^3b^n\! = \color{#0a0}{a^3} b^{3k-2} \equiv (b^3)^k \equiv 1\,\Rightarrow\, f_n \equiv 0$$
– user93957
Jan 3 '14 at 15:19
• $$\text{Guess who's back? Back again. Bill is back, tell a friend.}$$ I'm glad he's back. I've even made a silly souvenir. Jan 9 '14 at 15:45