Computing $\sqrt{2}$ with FRACTRAN

TL;DR

This paper introduces FRACTRAN programs that compute the decimal expansion of √2, demonstrating their relation to Conway's PIGAME and encoding the Newton-Raphson method within FRACTRAN for efficient computation.

Contribution

The paper presents new FRACTRAN programs for √2, provides a simpler proof of Conway's theorem, and encodes the Newton-Raphson method in FRACTRAN.

Findings

FRACTRAN programs for √2 are effective and conceptually linked to PIGAME.

A simplified proof of Conway's theorem is provided.

The Newton-Raphson method is encoded within FRACTRAN for √2 computation.

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.