## A Fractran Short Story

Fractran is a
Turing-complete esoteric programming language. A Fractran program is
just an ordered list of positive, irreducible fractions. The program's
output for an input *n* is the output of the program run
on *n* multiplied by the first fraction in the list that results
in an integer. If no such multiplication results in an integer, the
output is the input *n*. Variables are encoded in the exponents
of the prime factorization of the input and output.

Some time ago I thought up an idea for a short story involving
Fractran. A mathematician accidentally creates a Fractran program that
can trivially factor large composites. Think something like
O(log *n*). It's just the right magical string of, say, 31
fractions.

The story would be a first-person narrative of the mathematician's thoughts during a short time after the discovery, considering many of the consequences of the program. For example, it would render much of cryptography, which plays an essential role in the modern world, useless. He would also wonder if mankind should deserve such a discovery, considering how accidental it was.

This whole idea vanished once I realized that this Fractran program is actually completely trivial. It even runs in O(1) time. It's so trivial as to be worthless. Remember that Fractran stores its data in the number's prime factorization? The Fractran program that can factor any number in constant time is the identity function. To decode the output, which matches the input, all you need to do is factor it!

Interestingly, it doesn't seem to actually be possible to implement
the identity function in Fractran (*But somehow it's
Turing-complete? Hmmm... more investigation needed.*), unless you
can define your program in terms of its input. For example, the
program `1/(n+1)`

is the identity function for
input *n*.