# Congratulations, Bill! [duplicate]

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

## marked as duplicate by quid♦, Cookie, Martin Sleziak, Omnomnomnom, user642796Jan 17 '15 at 6:37

• 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$$ – user93957 Jan 3 '14 at 15:19
• I've deleted the comments about downvotes since such comments tend to become discussions about a users merit that do not really belong on a celebration thread. – Michael Greinecker Jan 4 '14 at 1:29
• $$\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. – Billy Rubina Jan 9 '14 at 15:45
• @GustavoBandeira: OMG, that's gold. :) – Prism Jan 13 '14 at 11:19
• Thanks! Congratulations and thanks to everyone for their efforts in helping to make MSE the best general-level math forum on the web. – Bill Dubuque Jan 17 '14 at 16:39