What are p-adic numbers ?
P-adic and rationnal numbers
P-adic numbers are an original way to look at the (limit of sequence of) elements in .
More precisely, just like represents the limits of Cauchy sequences in endowed with the distance : , represents the limits of Cauchy sequences in with another distance : , where is detailed below.
For a prime number, define as the exponent of in the decomposition of in a product of prime factors. Also define
Then is a distance on integers.
In with the distance , note that the sequence converges towards .
Why they matter
Various results can be proved using p-adic numbers. I discovered them in “Introduction to number theory”, where they are used to determine whether an ellipse has rationnal points. They also enable to give a meaning to
A p-adic number can be written where the sum might be infinite. Though it seems weird because the terms are growing, note that the sequence actually tends to really quickly in
A traditionnal way to picture p-adic numbers is with co-centric circles, like below:
All the credit goes to: Heiko Knopse for this illustration, more are available on his site
My idea is to take this idea to the limit. Formally, for , the complex number is associated to .
is a parameter between and used to ensure convergence.
Representing some integers
An interesting property is that . It is illustrated below. As you can see, addition in the p-addic representation shifts numbers to the right.
For those interested in number theory, I strongly recommend the following books, they are the reason I discovered p-adic integers and they motivated me to explore them (and write this article!)
Number Theory 1: Fermat’s Dream by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito
Number Theory 2: Introduction to Class Field Theory by the same authors which requires more knowledge in algebra and group theory.
“One square and an odd number of triangles”, a problem from Proofs from the book also makes an amazing use of p-adic valuations. The problem itself is simple to state:
is it possible to dissect a square into an odd number of triangles of equal area?
And this concept appears here, quite surprisingly.
from cmath import * class PAddicRepresenter: def __init__(self, p, l, output_length=30): self._p = p self._l = l self._output_length = output_length def to_plane(self, n): l = self._l p = self._p decomposed_int = self._completed_int_to_base(n) complex_coordinates = sum( [l ** n * exp(1j * c * 2 * pi / p) for n, c in enumerate(decomposed_int)]) return complex_coordinates.real, complex_coordinates.imag def transform_sample(self, ns): xs, ys = ,  for n in ns: x, y = self.to_plane(n) xs.append(x) ys.append(y) return xs, ys def _int_to_base(self, n): p = self._p i = 0 decomposition =  while n > 0: residual = n % p n = (n - residual) / p decomposition.append(residual) return decomposition def _completed_int_to_base(self, n): decomposed_int = self._int_to_base(n) return decomposed_int +  * (self._output_length - len(decomposed_int))
The first visualization being obtaining using the following:
import matplotlib.pyplot as plt plt.rcParams["figure.figsize"] = (8,8) from PAddicRepresenter import PAddicRepresenter n_points = 3**10 p = 3 small_sample_size = 55 l = 0.45 par = PAddicRepresenter(p, l) xs, ys = par.transform_sample(range(n_points)) fig, ax = plt.subplots() ax.hist2d(xs, ys, bins = 500, cmap = 'Greys') ax.scatter(xs[0:small_sample_size], ys[0:small_sample_size], c='black') for i in range(small_sample_size): ax.annotate(str(i), (xs[i] - 0.03 , ys[i] + 0.05)) plt.axis('off') plt.show()