What Do Numbers Look Like?

This article reproduces the code for johnhw’s essay What do numbers look like?

pip install umap-learn
1090.6s
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

    try: return totients[n]
    except KeyError: pass

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

    totients[n] = tot
    return 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('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=0.3, edgecolor='',
  c=np.arange(n), cmap="magma"
)

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