Quick Summary
FRACTRAN is an esolang built upon register machines, a theoretical alternative to turing machines for computation. We code the original 1987 paper to understand John Conway’s vision. We further code Geipel’s 2017 paper for fast cycle detection.
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As usual, you can code along to the GitHub.
The 1987 paper FRACTRAN: A Simple Universal Programming Language for Arithmetic (Conway, 1987)1 implements a register machine : an alternative to Turing machines where logic is represented within arithmetic, not as read/write heads on a tape (Wesstein, 2025)2.
In the simplest terms, Fractran is a computer architecture based entirely on the multiplication of fractions (Linvega, 2008)3.
The paper (Conway, 1987) asserts that a list of fractions alongside basic Start, Multiply, Stop rules permits Turing-complete computation.
Fractran uses prime numbers as registers and builds on the laws of logarithms we covered here. Here’s a quick recap for the noobs:
Fractan can be viewed as a slightly more general Turing machine in that it accepts an integer both as input and output, rather than outputting a boolean (Geipel, 2017)4.
So, using the laws of logarithms, Fractran accumulates results in the exponent of the prime factors of an integer.
A Fractran program is defined as a finite list of fractions. Given a positive integer input n, the algorithm calculates the next instruction by multiplying n with a fraction that yields another integer. The algorithm halts if no such fraction exists (Ramanathan, 2019)5.