I like Raganwald’s ode to Fractran which relates it to Minsky’s register machines and (surprisingly) the Collatz conjecture. https://raganwald.com/2020/05/03/fractran.html

This was a bit hard to read, requiring some note taking and some factorizating to check the correspondence between variable names and primes.

> You can get the product of two registers(x*y) by keeping an intermediate result and state register. Keeping the resulting product, by naming the first register for the result, prevents the accumulator grow too much in size:

    :: r acc x y
    :: iter acc > x iter
    :: iter >
    :: x y > r acc y 
    :: y > iter
    :: x > 

    AC 8575 x^2 y^2
    .. r^6

This one is harder to figure out. The first line reserves primes 2,3,5,7 for variables r, acc, x, y. The unreserved iter should then be assigned prime 11. The accumulator AC starts at value 8575 = 5^2 * 7^3, so the y^2 has a typo and should be y^3. Which matches the desired end result of 2*3 = 6. But how exactly does it get there?

Btw, the corresponding FRACTRAN program would be

    5*11  1   2*3*7  11  1
    ----  --  -----  --  -
    3*11  11   5*7    7  5

Some of the choices there make things harder to read.

For example, this machine takes two numbers and they are the numerator and denominator. Why on earth use ">" for the separator? That already has a mathematical meaning.

I think this site, along with the 100r.co sister site, are my favourite places on the Internet. Digital Garden indeed!

I think the Wikipedia page (https://en.wikipedia.org/wiki/FRACTRAN) is easier for understanding Fractan.

It reminds me of the https://en.wikipedia.org/wiki/Residue_number_system , it was actually used in a Russian computer system called "Duga" ("Curve") that needed fast multiplications.

It's still used for some niche purposes.

FRACTRAN interpreter in FRACTRAN, explaining how to write programs in FRACTRAN.

http://lomont.org/posts/2017/fractran/

Rational arithmetic machines have their uses. SAT solvers tend to have one inside. That's because rational arithmetic is closed under addition, subtraction, multiplication and division.

The reason to do this would to create a quantum computer emulator. I was looking into this myself recently. I believe Iran was doing the same things but on FPGAs:

https://www.tomshardware.com/news/iran-quantum-computer-arm-...