p-Adic Continued Fractions

numerator

38

denominator

40

prime

3

prime p = 3

19

20

=

-1+

1

-4/3+

1

-4/3+

1

-2/3+

1

-2/3+

1

2/3

The

p

-adic continued fraction

[

a

1

,

a

2

,…]

of a

p

-adic number is similar to the usual (simple) continued fraction in the reals with the requirement that

|

a

i

|≤p/2

. Since the rational numbers are a subset of the

p

-adics, every rational number has a unique

p

-adic continued fraction (which can be shown to be finite). This Demonstration computes the

p

-adic continued fractions for all rational numbers of the form

a/b

where

p

is less than 1000 and

a

and

b

are positive integers less than or equal to 100.