Computing $\sqrt{2}$ with FRACTRAN

Abstract

The FRACTRAN programs $\sqrt{2}$GAME and NR$\sqrt{2}$GAME are presented, both of which compute the decimal expansion of $\sqrt{2}$. Our $\sqrt{2}$GAME is analogous to Conway's PIGAME program. In fact, our proof carries over to PIGAME to produce a simpler proof of Conway's theorem as well as highlight how the efficiency of the program can be improved. NR$\sqrt{2}$GAME encodes the canonical example of the Newton--Raphson method in FRACTRAN.