The Unimaginable Mathematics of Borges' Library of Babel (13 page)

100 = 10
 10 = (2
 5)
 (2
 5) = (2
 2)
 (5
 5) = (2
²
)
 (5
²
)

and

100 = 4
 25 = (2
 2)
 (5
5) = (2
²
)
 (5
²
).

Because 100 is so familiar,
it's probably not surprising to you that both of the initial factorizations
lead to the unique one. And perhaps it is equally intuitive that no matter how
large an integer we begin with, no matter how we might try, there will be only
one way to factor it into powers of primes. Still, it's nice to know that
Euclid showed that it must always be true.

By the work
above, 5
^2,624,000
is a unique factorization of 25
^1,312,000
into
primes, each raised to a power greater than or equal to one. In this case,
plainly the number of distinct books uniquely decomposes to one prime (5)
raised to a power greater than one (2,624,000). It follows that the only
numbers that can divide 25
^1,312,000
are powers of five. Now, as is
easily inferred from the story, each hexagon in the Library contains 640 books.
The number 640 uniquely factors into 2
⁷
5, and so the number 640 does not divide 25
^1,312,000
,
for