FRACTRAN
FRACTRAN is an esoteric programming language developed by John Conway that is based solely on fractions and prime numbers.
Each program consists of a starting number as input (n), acting as an accumulator does in lowlevel assembly languages, as it persists and changes throughout the program. The instructions consist of a series of fractions (each fraction being represented as f) that are cycled through using the following rules[1,4]:
 for the first fraction f in the list [of fractions] for which [the result of] nf is an integer, replace n by nf
 repeat this rule until no fraction f in the list produces an integer when multiplied by n, then halt.
Example
We will take the most simple example, an adding machine, to go through the rules. Don't worry if it doesn't make sense yet, this is one of those things that is really hard to understand without going through the whole thing first.
The instruction set for the adding machine is a single instruction:
2/3
Following rule 1, with an input (n) of 18
aka 2^1 * 3^2
and the list of fractions (f) being a single fraction of 2/3
:
 We go through each fraction lefttoright, determining whether nf would be an integer. Looking at the first fraction, the result would be an integer:
18 * 2 / 3
, or12
aka2^2 * 3^1
. Having found "the first fraction f in the list [of fractions] for which [the result of] nf is an integer", we "replace n by nf" and repeat again.  We repeat rule 1 with the new n of
12
aka2^2 * 3^1
, going through each fraction f lefttoright, determining whether nf would be an integer. We look at the first fraction f and see the result would again resolve into an integer:12 * 2 / 3
, or8
, aka2^3
. Again we "replace n by nf" and repeat.  Repeating rule 1 with the new n of
8
aka2^3
, we look at the first fraction f and see the result would not resolve into an integer:2 * 8 / 3
or16 / 3
. If we had another fraction f in our list of fractions, we would continue to the next one and test, repeating the above process. But because we have "[repeated] this rule until no fraction f in the list produces an integer when multiplied by n", we halt the program.  Our resulting n is
8
, or2^3
.
How It Works
Going through this section, I highly recommend going over the above example between each section, integrating the new ideas to better understand what is happening. Going through the paper computing examples on this site will also prepare you for this, if you aren't too familiar with lowlevel programming.
Memory and Registers
Similar to lowlevel languages like assembly, we maintain and operate on numbers in discrete places in memory (or registers). However, these pieces of memory or registers are all contained within our number n by way of being a composite number composed of prime numbers. Therefore, our memory or registers are represented as powers of prime numbers.
Lets use our starting number above, 18
. This can be reduced down to 2 * 3 * 3
, or 2^1 * 3^2
. The number 18
is viewed in FRACTRAN as representing the "2 register" having a value of 1
, and the "3 register" having a value of 2
. Similarly, the number 120
can be reduced to 2^3 * 3^1 * 5^1
, which would have register values as follows:
Register  Value 

r2  3 
r3  1 
r5  1 
r7  0 
...  ... 
So instead of having named variables like X
, Y
, or counter
and altering them directly, we store our mutable data in these registers and operate on them using instructions.
Instructions
There are only two types of instructions: incrementing or decrementing a register.
In our list of fractions, we can view each numerator/denominator pair as follows:
if condition in denominator is met,
numerator then increment {numerator primes at their powers}
 = 
denominator if n contains {denominator primes at their powers},
then decrement {denominator primes at their powers}
So again, lets use our simple adding machine as the example, particularly the fraction 2/3
. First, we need to reduce the numerator and denominator to its composite parts:
2 2^1
 = 
3 3^1
Then we can write out the step in plain English:
If the composition of n contains {at least 1 in register 3}, then
decrement {1 in register 3} AND
increment {1 in register 2}.
Or in a more discrete and programish way:
if (r3 >= 1) {
r3 = 1;
r2 += 1;
}
Using a more extreme example, we could use a fraction like 365,625 / 242
, or 3^2 * 5^5 * 13^1 / 2^1 * 11^2
. This could be written as:
if (r2 >= 1 && r11 >= 2) {
r2 = 1;
r11 = 2;
r3 += 2;
r5 += 5;
r13 += 1;
}
Axioms
These are some things I've deduced after messing with this and seeing some programs.

Numerators increment the primes that are present there.

Denominators act as an if/elseif statement and decrement the primes that are present there.

Functions can be thought of as a giant while loop with a single if/elseif chain inside. e.g. the pointless program
(2/3, 5/4)
would look like:while true { if (r3 >= 1) { r3 = 1; r2 += 1; } else if (r2 >= 2) { r2 = 2; r5 += 1; } else { break; } }

A fraction with a denominator of 1 means the program will not ever go beyond it, as it will always be true. For example, the final fraction in PRIMEGAME[6] is
55/1
, as it should repeatedly put out prime numbers forever. 
A fraction with a numerator of 1 means that the number in the denominator's register(s) will need to be zero before it ever reaches a fraction beyond it. For instance, in the multiplication program[7], the fraction
1/11
defines the end of a certain state; to get past that fraction and to the other state, the11
register needs to be depleted to11^0
. 
It is not possible for a FRACTRAN denominator to directly test if a variable is 0 (However, an indirect test can be implemented by creating a default fraction that is placed after other fractions that test a particular variable.).[1]
Examples[10]
You can find a lot more details on each of these in the references below.
Comparators
Greater Than
INPUT (n): 2^a * 3^b * 5^isGreater
OUTPUT: 5^isGreater
r2: a // first number
r3: b // second number
r5: isGreater // boolean; starts off at 1
(
1 / 6,
1 / 15,
1 / 2,
1 / 3
)
Less Than
INPUT (n): 2^a * 3^b * 5^isLess
OUTPUT: 5^isLess
r2: a // first number
r3: b // second number
r5: isLess // boolean; starts off at 1
(
1 / 6,
1 / 10,
1 / 2,
1 / 3
)
Equal To
INPUT (n): 2^a * 3^b + 5^isEqual
OUTPUT: 5^isEqual
r2: a // first number
r3: b // second number
r5: isEqual // boolean; starts off at 1
(
1 / 2 * 3,
1 / 2 * 5,
1 / 3 * 5,
1 / 2,
1 / 3
)
Subtraction
INPUT (n): 2^a * 3^b
OUTPUT: 2^a  b (or) 3^b  a
r2: a // first number
r3: b // second number
(
1 / 2 * 3
)
Multiplication
INPUT (n): 2^a * 3^b
OUTPUT: 5^a * b
r2: a // first number
r3: b // second number
r5: Product // resulting product
r7: Temp // aids in the repeated addition
r11,r13: State Beta One and Two // changes to restart addition process
(
5 * 7 * 13 / 3 * 11,
11 / 13,
1 / 11,
3 / 7,
11 / 2,
1 / 3,
)
Division
INPUT (n): 2^a * 3^b
OUTPUT: 5^a / b, 7^a % b
r2: Dividend
r3: Divisor
r5: Quotient
r7: Remainder
r11r13: State Alpha One and Two // starts with Alpha One at 1
r17r19: State Beta One and Two
(
7 * 13 / 2 * 3 * 11,
11 / 13,
1 / 3 * 11,
5 * 17 / 11,
3 * 19 / 7 * 17,
17 / 19,
11 / 17,
1 / 3
)
Euler Problem 001
Honestly, this is terrible, but it does work. The answer comes out to 7^233,168, which is about 10^(10^5.29)
INPUT (n): 13^State Alpha One
OUTPUT: 7^sum
r2: total
r3: three
r5: five
r7: sum
r11: temp
r13,r17: State Alpha One and Two
r19,r23,r29: State Beta Three, Five, and ThreeAndFive
r31,r37: State Gamma One and Two
r41,r43: State Delta One and Two
r47: end
(
47 / 2^1000 * 13,
17 / 3 * 5 * 13,
13 / 17,
19 / 3 * 13,
23 / 5 * 13,
29 / 13,
3 * 5^5 * 31 / 19,
3^3 * 5 * 31 / 23,
3^3 * 5^5 * 31 / 29,
7 * 11 * 37 / 2 * 31,
31 / 37,
41 / 31,
2 * 43 / 11 * 41,
41 / 43,
13 / 41
)
References
 https://en.wikipedia.org/wiki/FRACTRAN
 https://wiki.xxiivv.com/site/fractran.html
 https://www.youtube.com/watch?v=548BHYFT1E
 John Conway's original paper
 https://esolangs.org/wiki/Fractran
 https://link.springer.com/chapter/10.1007%2F9781461248088_2
 https://en.wikipedia.org/wiki/FRACTRAN#Multiplication
 http://raganwald.com/2020/05/03/fractran.html
 https://github.com/raganwald/FRACTRAN
 Example Programs
Incoming Links
Last modified: 202401040446