
Recounting the rationals (2008) - godelmachine
http://fermatslibrary.com/s/recounting-the-rationals
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aisopous
I find this incredibly beautiful. Short papers that introduce a "general"
audience to a supposedly well-known mathematical idea should really be seen as
a valuable contribution in mathematics.

Granted, I am a pure maths PhD. I would be really curious to know how many
non-mathematicians can glean value/enjoyment from reading this paper, and what
if anything could constitute a necessary/sufficient condition for achieving
that.

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szemet
I still find that the sortest one is this:

take the prime factors of _n_ , the primes with even power go to the numerator
with halved powers, the primes with odd powers go to the denominator also with
halved powers but rounded up. This is bijective, due to the uniqueness of
prime factorization, and of course the numerator and denominator greatest
divisor is also guaranteed to be 1, as they not share prime factors at all. So
e.g.

45 = 3² _5¹- > 3/5 60 = 2²_3¹*5¹ -> 2/15 2 -> 1/2, 3 -> 1/3, 4 -> 2/1, 5 ->
1/5, 6 -> 1/6, 7 -> 1/7, 8 -> 1/4, 9 -> 3/1, ...

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saywatnow
pdf:
[https://www.math.upenn.edu/~wilf/website/recounting.pdf](https://www.math.upenn.edu/~wilf/website/recounting.pdf)

