Simon Danisch / Dec 05 2018

# What Do Numbers Look Like?

This article reproduces the code from johnhw’s essay What Do Numbers Look Like.

pip install umap-learn
16.9s
Language:Python
### JHW 2018
import numpy as np
import umap

# This code from the excellent module at:
# https://stackoverflow.com/questions/4643647/fast-prime-factorization-module

import random

_known_factors = {}

totients = {}
def primesbelow(N):
# http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
#""" Input N>=6, Returns a list of primes, 2 <= p < N """
correction = N % 6 > 1
N = {0:N, 1:N-1, 2:N+4, 3:N+3, 4:N+2, 5:N+1}[N%6]
sieve = [True] * (N // 3)
sieve[0] = False
for i in range(int(N ** .5) // 3 + 1):
if sieve[i]:
k = (3 * i + 1) | 1
sieve[k*k // 3::2*k] = [False] * ((N//6 - (k*k)//6 - 1)//k + 1)
sieve[(k*k + 4*k - 2*k*(i%2)) // 3::2*k] = [False] * ((N // 6 - (k*k + 4*k - 2*k*(i%2))//6 - 1) // k + 1)
return [2, 3] + [(3 * i + 1) | 1 for i in range(1, N//3 - correction) if sieve[i]]

N = 1000000
smallprimeset = set(primesbelow(N))
_smallprimeset = N

smallprimes = primesbelow(10000000)
prime_ix = {p:i for i,p in enumerate(smallprimes)}

def isprime(n, precision=7):
# http://en.wikipedia.org/wiki/Miller-Rabin_primality_test#Algorithm_and_running_time
if n < 1:
raise ValueError("Out of bounds, first argument must be > 0")
elif n <= 3:
return n >= 2
elif n % 2 == 0:
return False
elif n < _smallprimeset:
return n in smallprimeset

d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1

for repeat in range(precision):
a = random.randrange(2, n - 2)
x = pow(a, d, n)

if x == 1 or x == n - 1: continue

for r in range(s - 1):
x = pow(x, 2, n)
if x == 1: return False
if x == n - 1: break
else: return False

return True

# https://comeoncodeon.wordpress.com/2010/09/18/pollard-rho-brent-integer-factorization/
def pollard_brent(n):
if n % 2 == 0: return 2
if n % 3 == 0: return 3

y, c, m = random.randint(1, n-1), random.randint(1, n-1), random.randint(1, n-1)
g, r, q = 1, 1, 1
while g == 1:
x = y
for i in range(r):
y = (pow(y, 2, n) + c) % n

k = 0
while k < r and g==1:
ys = y
for i in range(min(m, r-k)):
y = (pow(y, 2, n) + c) % n
q = q * abs(x-y) % n
g = gcd(q, n)
k += m
r *= 2
if g == n:
while True:
ys = (pow(ys, 2, n) + c) % n
g = gcd(abs(x - ys), n)
if g > 1:
break

return g

def _primefactors(n, sort=False):
factors = []

for checker in smallprimes:
while n % checker == 0:
factors.append(checker)
n //= checker
# early exit memoization
if n in _known_factors:
return factors + _known_factors[n]
if checker > n: break

if n < 2: return factors

while n > 1:
if isprime(n):
factors.append(n)
break
factor = pollard_brent(n) # trial division did not fully factor, switch to pollard-brent
factors.extend(primefactors(factor)) # recurse to factor the not necessarily prime factor returned by pollard-brent
n //= factor

if sort: factors.sort()

return factors

def primefactors(n, sort=False):
if n in _known_factors:
return _known_factors[n]

result = _primefactors(n)
_known_factors[n] = result
return result

from collections import defaultdict

def factorization(n):
factors = defaultdict(int)
for p1 in primefactors(n):
factors[p1] += 1
return factors

def unique_factorise(n):
return set(primefactors(n))

def totient(n):
if n == 0: return 1

except KeyError: pass

tot = 1
for p, exp in factorization(n).items():
tot *= (p - 1)  *  p ** (exp - 1)

totients[n] = tot

def gcd(a, b):
if a == b: return a
while b > 0: a, b = b, a % b
return a

def lcm(a, b):
return abs((a // gcd(a, b)) * b)

### end

## Create sparse binary factor vectors for any number, and assemble into a matrix
## One column for each unique prime factor
## One row for each number, 0=does not have this factor, 1=does have this factor (might be repeated)

from scipy.special import expi
import scipy.sparse

def factor_vector_lil(n):
## approximate prime counting function (upper bound for the values we are interested in)
## gives us the number of rows (dimension of our space)
d = int(np.ceil(expi(np.log(n))))
x = scipy.sparse.lil_matrix((n,d))
for i in range(2,n):
for k,v in factorization(i).items():
x[i,prime_ix[k]] = 1

if i%100000==0: # just check it is still alive...
print(i)
return x

### Generate the matrix for 1 million integers

n = 100000
X = factor_vector_lil(n)

# embed with UMAP
embedding = umap.UMAP(metric='cosine', n_epochs=500).fit_transform(X)

# save for later
np.savez('/results/1e6_pts.npz', embedding=embedding);
# and save the image
from matplotlib import pyplot as plt
fig = plt.figure(figsize=(8,8))
fig.patch.set_facecolor('black')
plt.scatter(
embedding[:,0], embedding[:,1], marker='o', s=1, edgecolor='',
c=np.arange(n), cmap="magma"
)

plt.axis("off")
plt.savefig("results/primes_umap_1e6_16k_smaller_pts.png", dpi=100, facecolor='black')