O que é uma casa prime?

Por exemplo, considere HP (4). Primeiro, encontre os fatores principais. Os fatores primos de 4 ( em ordem numérica do menor para o maior, sempre ) são 2, 2. Considere esses fatores como um número literal. 2, 2 se torna 22. Esse processo de fatoração continua até você atingir um número primo.

number    prime factors
4         2, 2
22        2, 11
211       211 is prime

Quando você alcança um número primo, a sequência termina. HP (4) = 211. Aqui está um exemplo mais longo, com 14:

number    prime factors
14        2, 7
27        3, 3, 3
333       3, 3, 37
3337      47, 71
4771      13, 367
13367     13367 is prime

Seu desafio é criar um programa que calcule HP (x) dado x, e faça-o o mais rápido possível . Você pode usar quaisquer recursos que desejar, exceto uma lista de primos iniciais conhecidos.

Tome nota, esses números se tornam muito grandes muito rapidamente. Em x = 8, o HP (x) salta até 3331113965338635107. O HP (49) ainda não foi encontrado.

A velocidade do programa será testada em um Raspberry Pi 2, com a média das seguintes entradas:

16
20
64
65
80

Se você possui um Raspberry Pi 2, programe o programa por conta própria e calcule a média dos tempos.

Mathematica, HP (80) em ~ 0.88s

NestWhile[
  FromDigits[
    Flatten[IntegerDigits /@ 
      ConstantArray @@@ FactorInteger[#]]] &, #, CompositeQ] &

Função anônima. Pega um número como entrada e retorna um número como saída.

PyPy 5.4.1 64 bits (linux), HP (80) ~ 1.54s

A versão de 32 bits terá um tempo um pouco mais lento.

Eu uso quatro métodos diferentes de fatoração, com pontos de interrupção empiricamente determinados:

• n <2 ³⁰ → fatoração por divisão de teste
• n <2 ⁷⁰ → Rho de Pollard (variante de Brent)
• else → Lenstra ECF (usando curvas de Montgomery, parâmetros de Suyama), teste ~ 100 curvas
• se o ECF falhou → Peneira quadrática polinomial múltipla (usando polinômios de Montgomery)

Tentei por um tempo encontrar uma pausa limpa entre ECF e MPQS, mas parece não haver uma. No entanto, se n contiver um fator pequeno, o ECF geralmente o encontrará quase imediatamente, então optei por testar apenas algumas curvas, antes de passar para o MPQS.

Atualmente, é apenas duas vezes mais lento que o Mathmatica, o que certamente considero um sucesso.

home-prime.py

import math
import my_math
import mpqs

max_trial = 1e10
max_pollard = 1e22

def factor(n):
  if n < max_trial:
    return factor_trial(n)
  for p in my_math.small_primes:
    if n%p == 0:
      return [p] + factor(n/p)
  if my_math.is_prime(n):
    return [n]
  if n < max_pollard:
    p = pollard_rho(n)
  else:
    p = lenstra_ecf(n) or mpqs.mpqs(n)
  return factor(p) + factor(n/p)


def factor_trial(n):
  a = []
  for p in my_math.small_primes:
    while n%p == 0:
      a += [p]
      n /= p
  i = 211
  while i*i < n:
    for o in my_math.offsets:
      i += o
      while n%i == 0:
        a += [i]
        n /= i
  if n > 1:
    a += [n]
  return a


def pollard_rho(n):
  # Brent's variant
  y, r, q = 0, 1, 1
  c, m = 9, 40
  g = 1
  while g == 1:
    x = y
    for i in range(r):
      y = (y*y + c) % n
    k = 0
    while k < r and g == 1:
      ys = y
      for j in range(min(m, r-k)):
        y = (y*y + c) % n
        q = q*abs(x-y) % n
      g = my_math.gcd(q, n)
      k += m
    r *= 2
  if g == n:
    ys = (ys*ys + c) % n
    g = gcd(n, abs(x-ys))
    while g == 1:
      ys = (ys*ys + c) % n
      g = gcd(n, abs(x-ys))
  return g

def ec_add((x1, z1), (x2, z2), (x0, z0), n):
  t1, t2 = (x1-z1)*(x2+z2), (x1+z1)*(x2-z2)
  x, z = t1+t2, t1-t2
  return (z0*x*x % n, x0*z*z % n)

def ec_double((x, z), (a, b), n):
  t1 = x+z; t1 *= t1
  t2 = x-z; t2 *= t2
  t3 = t1 - t2
  t4 = 4*b*t2
  return (t1*t4 % n, t3*(t4 + a*t3) % n)

def ec_multiply(k, p, C, n):
  # Montgomery ladder algorithm
  p0 = p
  q, p = p, ec_double(p, C, n)
  b = k >> 1
  while b > (b & -b):
    b ^= b & -b
  while b:
    if k&b:
      q, p = ec_add(p, q, p0, n), ec_double(p, C, n)
    else:
      q, p = ec_double(q, C, n), ec_add(p, q, p0, n),
    b >>= 1
  return q

def lenstra_ecf(n, m = 5):
  # Montgomery curves w/ Suyama parameterization.
  # Based on pseudocode found in:
  # "Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware"
  # Gaj, Kris et. al
  # http://www.hyperelliptic.org/tanja/SHARCS/talks06/Gaj.pdf
  # Phase 2 is not implemented.
  B1, B2 = 8, 13
  for i in range(m):
    pg = my_math.primes()
    p = pg.next()
    k = 1
    while p < B1:
      k *= p**int(math.log(B1, p))
      p = pg.next()
    for s in range(B1, B2):
      u, v = s*s-5, 4*s
      x = u*u*u
      z = v*v*v
      t = pow(v-u, 3, n)
      P = (x, z)
      C = (t*(3*u+v) % n, 4*x*v % n)
      Q = ec_multiply(k, P, C, n)
      g = my_math.gcd(Q[1], n)
      if 1 < g < n: return g
    B1, B2 = B2, B1 + B2


if __name__ == '__main__':
  import time
  import sys
  for n in sys.argv[1:]:
    t0 = time.time()
    i = int(n)
    f = []
    while len(f) != 1:
      f = sorted(factor(i))
      #print i, f
      i = int(''.join(map(str, f)))
    t1 = time.time()-t0
    print n, i
    print '%.3fs'%(t1)
    print

Exemplo de tempos

    $ pypy home-prime.py 8 16 20 64 65 80
8 3331113965338635107
0.005s

16 31636373
0.001s

20 3318308475676071413
0.004s

64 1272505013723
0.000s

65 1381321118321175157763339900357651
0.397s

80 313169138727147145210044974146858220729781791489
1.537s

A média dos 5 é de cerca de 0,39s.

Dependências

mpqs.pyé extraído diretamente da minha resposta à fatoração semiprime mais rápida, com algumas modificações muito pequenas.

mpqs.py

import math
import my_math
import time

# Multiple Polynomial Quadratic Sieve
def mpqs(n, verbose=False):
  if verbose:
    time1 = time.time()

  root_n = my_math.isqrt(n)
  root_2n = my_math.isqrt(n+n)

  # formula chosen by experimentation
  # seems to be close to optimal for n < 10^50
  bound = int(5 * math.log(n, 10)**2)

  prime = []
  mod_root = []
  log_p = []
  num_prime = 0

  # find a number of small primes for which n is a quadratic residue
  p = 2
  while p < bound or num_prime < 3:

    # legendre (n|p) is only defined for odd p
    if p > 2:
      leg = my_math.legendre(n, p)
    else:
      leg = n & 1

    if leg == 1:
      prime += [p]
      mod_root += [int(my_math.mod_sqrt(n, p))]
      log_p += [math.log(p, 10)]
      num_prime += 1
    elif leg == 0:
      if verbose:
        print 'trial division found factors:'
        print p, 'x', n/p
      return p

    p = my_math.next_prime(p)

  # size of the sieve
  x_max = bound*8

  # maximum value on the sieved range
  m_val = (x_max * root_2n) >> 1

  # fudging the threshold down a bit makes it easier to find powers of primes as factors
  # as well as partial-partial relationships, but it also makes the smoothness check slower.
  # there's a happy medium somewhere, depending on how efficient the smoothness check is
  thresh = math.log(m_val, 10) * 0.735

  # skip small primes. they contribute very little to the log sum
  # and add a lot of unnecessary entries to the table
  # instead, fudge the threshold down a bit, assuming ~1/4 of them pass
  min_prime = int(thresh*3)
  fudge = sum(log_p[i] for i,p in enumerate(prime) if p < min_prime)/4
  thresh -= fudge

  sieve_primes = [p for p in prime if p >= min_prime]
  sp_idx = prime.index(sieve_primes[0])

  if verbose:
    print 'smoothness bound:', bound
    print 'sieve size:', x_max
    print 'log threshold:', thresh
    print 'skipping primes less than:', min_prime

  smooth = []
  used_prime = set()
  partial = {}
  num_smooth = 0
  prev_num_smooth = 0
  num_used_prime = 0
  num_partial = 0
  num_poly = 0
  root_A = my_math.isqrt(root_2n / x_max)

  if verbose:
    print 'sieving for smooths...'
  while True:
    # find an integer value A such that:
    # A is =~ sqrt(2*n) / x_max
    # A is a perfect square
    # sqrt(A) is prime, and n is a quadratic residue mod sqrt(A)
    while True:
      root_A = my_math.next_prime(root_A)
      leg = my_math.legendre(n, root_A)
      if leg == 1:
        break
      elif leg == 0:
        if verbose:
          print 'dumb luck found factors:'
          print root_A, 'x', n/root_A
        return root_A

    A = root_A * root_A

    # solve for an adequate B
    # B*B is a quadratic residue mod n, such that B*B-A*C = n
    # this is unsolvable if n is not a quadratic residue mod sqrt(A)
    b = my_math.mod_sqrt(n, root_A)
    B = (b + (n - b*b) * my_math.mod_inv(b + b, root_A))%A

    # B*B-A*C = n <=> C = (B*B-n)/A
    C = (B*B - n) / A

    num_poly += 1

    # sieve for prime factors
    sums = [0.0]*(2*x_max)
    i = sp_idx
    for p in sieve_primes:
      logp = log_p[i]

      inv_A = my_math.mod_inv(A, p)
      # modular root of the quadratic
      a = int(((mod_root[i] - B) * inv_A)%p)
      b = int(((p - mod_root[i] - B) * inv_A)%p)

      amx = a+x_max
      bmx = b+x_max

      ax = amx-p
      bx = bmx-p

      k = p
      while k < x_max:
        sums[k+ax] += logp
        sums[k+bx] += logp
        sums[amx-k] += logp
        sums[bmx-k] += logp
        k += p

      if k+ax < x_max:  
        sums[k+ax] += logp
      if k+bx < x_max:
        sums[k+bx] += logp
      if amx-k > 0:
        sums[amx-k] += logp
      if bmx-k > 0:
        sums[bmx-k] += logp
      i += 1

    # check for smooths
    x = -x_max
    for v in sums:
      if v > thresh:
        vec = set()
        sqr = []
        # because B*B-n = A*C
        # (A*x+B)^2 - n = A*A*x*x+2*A*B*x + B*B - n
        #               = A*(A*x*x+2*B*x+C)
        # gives the congruency
        # (A*x+B)^2 = A*(A*x*x+2*B*x+C) (mod n)
        # because A is chosen to be square, it doesn't need to be sieved
        sieve_val = (A*x + B+B)*x + C

        if sieve_val < 0:
          vec = {-1}
          sieve_val = -sieve_val

        for p in prime:
          while sieve_val%p == 0:
            if p in vec:
              # keep track of perfect square factors
              # to avoid taking the sqrt of a gigantic number at the end
              sqr += [p]
            vec ^= {p}
            sieve_val = int(sieve_val / p)

        if sieve_val == 1:
          # smooth
          smooth += [(vec, (sqr, (A*x+B), root_A))]
          used_prime |= vec
        elif sieve_val in partial:
          # combine two partials to make a (xor) smooth
          # that is, every prime factor with an odd power is in our factor base
          pair_vec, pair_vals = partial[sieve_val]
          sqr += list(vec & pair_vec) + [sieve_val]
          vec ^= pair_vec
          smooth += [(vec, (sqr + pair_vals[0], (A*x+B)*pair_vals[1], root_A*pair_vals[2]))]
          used_prime |= vec
          num_partial += 1
        else:
          # save partial for later pairing
          partial[sieve_val] = (vec, (sqr, A*x+B, root_A))
      x += 1

    prev_num_smooth = num_smooth
    num_smooth = len(smooth)
    num_used_prime = len(used_prime)

    if verbose:
      print 100 * num_smooth / num_prime, 'percent complete\r',

    if num_smooth > num_used_prime and num_smooth > prev_num_smooth:
      if verbose:
        print '%d polynomials sieved (%d values)'%(num_poly, num_poly*x_max*2)
        print 'found %d smooths (%d from partials) in %f seconds'%(num_smooth, num_partial, time.time()-time1)
        print 'solving for non-trivial congruencies...'

      used_prime_list = sorted(list(used_prime))

      # set up bit fields for gaussian elimination
      masks = []
      mask = 1
      bit_fields = [0]*num_used_prime
      for vec, vals in smooth:
        masks += [mask]
        i = 0
        for p in used_prime_list:
          if p in vec: bit_fields[i] |= mask
          i += 1
        mask <<= 1

      # row echelon form
      col_offset = 0
      null_cols = []
      for col in xrange(num_smooth):
        pivot = col-col_offset == num_used_prime or bit_fields[col-col_offset] & masks[col] == 0
        for row in xrange(col+1-col_offset, num_used_prime):
          if bit_fields[row] & masks[col]:
            if pivot:
              bit_fields[col-col_offset], bit_fields[row] = bit_fields[row], bit_fields[col-col_offset]
              pivot = False
            else:
              bit_fields[row] ^= bit_fields[col-col_offset]
        if pivot:
          null_cols += [col]
          col_offset += 1

      # reduced row echelon form
      for row in xrange(num_used_prime):
        # lowest set bit
        mask = bit_fields[row] & -bit_fields[row]
        for up_row in xrange(row):
          if bit_fields[up_row] & mask:
            bit_fields[up_row] ^= bit_fields[row]

      # check for non-trivial congruencies
      for col in null_cols:
        all_vec, (lh, rh, rA) = smooth[col]
        lhs = lh   # sieved values (left hand side)
        rhs = [rh] # sieved values - n (right hand side)
        rAs = [rA] # root_As (cofactor of lhs)
        i = 0
        for field in bit_fields:
          if field & masks[col]:
            vec, (lh, rh, rA) = smooth[i]
            lhs += list(all_vec & vec) + lh
            all_vec ^= vec
            rhs += [rh]
            rAs += [rA]
          i += 1

        factor = my_math.gcd(my_math.list_prod(rAs)*my_math.list_prod(lhs) - my_math.list_prod(rhs), n)
        if 1 < factor < n:
          break
      else:
        if verbose:
          print 'none found.'
        continue
      break

  if verbose:
    print 'factors found:'
    print factor, 'x', n/factor
    print 'time elapsed: %f seconds'%(time.time()-time1)
  return factor

if __name__ == "__main__":
  import argparse
  parser = argparse.ArgumentParser(description='Uses a MPQS to factor a composite number')
  parser.add_argument('composite', metavar='number_to_factor', type=long, help='the composite number to factor')
  parser.add_argument('--verbose', dest='verbose', action='store_true', help="enable verbose output")
  args = parser.parse_args()

  if args.verbose:
    mpqs(args.composite, args.verbose)
  else:
    time1 = time.time()
    print mpqs(args.composite)
    print 'time elapsed: %f seconds'%(time.time()-time1)

my_math.pyé retirado do mesmo post que mpqs.py, no entanto, também adicionei o gerador de número primo ilimitado que usei na minha resposta para encontrar a maior lacuna entre os primos bons .

my_math.py

# primes less than 212
small_primes = [
    2,  3,  5,  7, 11, 13, 17, 19, 23, 29, 31, 37,
   41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,
   97,101,103,107,109,113,127,131,137,139,149,151,
  157,163,167,173,179,181,191,193,197,199,211]

# pre-calced sieve of eratosthenes for n = 2, 3, 5, 7
indices = [
    1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
   53, 59, 61, 67, 71, 73, 79, 83, 89, 97,101,103,
  107,109,113,121,127,131,137,139,143,149,151,157,
  163,167,169,173,179,181,187,191,193,197,199,209]

# distances between sieve values
offsets = [
  10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6,
   6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4,
   2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6,
   4, 2, 4, 6, 2, 6, 4, 2, 4, 2,10, 2]

# tabulated, mod 105
dindices =[
  0,10, 2, 0, 4, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0,
  0, 6, 2, 0, 4, 0, 0, 4, 6, 0, 0, 6, 0, 0, 2,
  0, 6, 2, 0, 4, 0, 0, 4, 6, 0, 0, 2, 0, 4, 2,
  0, 6, 6, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 4, 2,
  0, 6, 2, 0, 4, 0, 0, 4, 6, 0, 0, 2, 0, 6, 2,
  0, 6, 0, 0, 4, 0, 0, 4, 6, 0, 0, 2, 0, 4, 8,
  0, 0, 2, 0,10, 0, 0, 4, 0, 0, 0, 2, 0, 4, 2]

max_int = 2147483647


# returns the index of x in a sorted list a
# or the index of the next larger item if x is not present
# i.e. the proper insertion point for x in a
def binary_search(a, x):
  s = 0
  e = len(a)
  m = e >> 1
  while m != e:
    if a[m] < x:
      s = m
      m = (s + e + 1) >> 1
    else:
      e = m
      m = (s + e) >> 1
  return m


# divide and conquer list product
def list_prod(a):
  size = len(a)
  if size == 1:
    return a[0]
  return list_prod(a[:size>>1]) * list_prod(a[size>>1:])


# greatest common divisor of a and b
def gcd(a, b):
  while b:
    a, b = b, a%b
  return a


# extended gcd
def ext_gcd(a, m):
  a = int(a%m)
  x, u = 0, 1
  while a:
    x, u = u, x - (m/a)*u
    m, a = a, m%a
  return (m, x, u)


# legendre symbol (a|m)
# note: returns m-1 if a is a non-residue, instead of -1
def legendre(a, m):
  return pow(a, (m-1) >> 1, m)


# modular inverse of a mod m
def mod_inv(a, m):
  return ext_gcd(a, m)[1]


# modular sqrt(n) mod p
# p must be prime
def mod_sqrt(n, p):
  a = n%p
  if p%4 == 3:
    return pow(a, (p+1) >> 2, p)
  elif p%8 == 5:
    v = pow(a << 1, (p-5) >> 3, p)
    i = ((a*v*v << 1) % p) - 1
    return (a*v*i)%p
  elif p%8 == 1:
    # Shank's method
    q = p-1
    e = 0
    while q&1 == 0:
      e += 1
      q >>= 1

    n = 2
    while legendre(n, p) != p-1:
      n += 1

    w = pow(a, q, p)
    x = pow(a, (q+1) >> 1, p)
    y = pow(n, q, p)
    r = e
    while True:
      if w == 1:
        return x

      v = w
      k = 0
      while v != 1 and k+1 < r:
        v = (v*v)%p
        k += 1

      if k == 0:
        return x

      d = pow(y, 1 << (r-k-1), p)
      x = (x*d)%p
      y = (d*d)%p
      w = (w*y)%p
      r = k
  else: # p == 2
    return a


#integer sqrt of n
def isqrt(n):
  c = n*4/3
  d = c.bit_length()

  a = d>>1
  if d&1:
    x = 1 << a
    y = (x + (n >> a)) >> 1
  else:
    x = (3 << a) >> 2
    y = (x + (c >> a)) >> 1

  if x != y:
    x = y
    y = (x + n/x) >> 1
    while y < x:
      x = y
      y = (x + n/x) >> 1
  return x


# integer cbrt of n
def icbrt(n):
  d = n.bit_length()

  if d%3 == 2:
    x = 3 << d/3-1
  else:
    x = 1 << d/3

  y = (2*x + n/(x*x))/3
  if x != y:
    x = y
    y = (2*x + n/(x*x))/3
    while y < x:
      x = y
      y = (2*x + n/(x*x))/3
  return x


# strong probable prime
def is_sprp(n, b=2):
  if n < 2: return False
  d = n-1
  s = 0
  while d&1 == 0:
    s += 1
    d >>= 1

  x = pow(b, d, n)
  if x == 1 or x == n-1:
    return True

  for r in xrange(1, s):
    x = (x * x)%n
    if x == 1:
      return False
    elif x == n-1:
      return True

  return False


# lucas probable prime
# assumes D = 1 (mod 4), (D|n) = -1
def is_lucas_prp(n, D):
  P = 1
  Q = (1-D) >> 2

  # n+1 = 2**r*s where s is odd
  s = n+1
  r = 0
  while s&1 == 0:
    r += 1
    s >>= 1

  # calculate the bit reversal of (odd) s
  # e.g. 19 (10011) <=> 25 (11001)
  t = 0
  while s:
    if s&1:
      t += 1
      s -= 1
    else:
      t <<= 1
      s >>= 1

  # use the same bit reversal process to calculate the sth Lucas number
  # keep track of q = Q**n as we go
  U = 0
  V = 2
  q = 1
  # mod_inv(2, n)
  inv_2 = (n+1) >> 1
  while t:
    if t&1:
      # U, V of n+1
      U, V = ((U + V) * inv_2)%n, ((D*U + V) * inv_2)%n
      q = (q * Q)%n
      t -= 1
    else:
      # U, V of n*2
      U, V = (U * V)%n, (V * V - 2 * q)%n
      q = (q * q)%n
      t >>= 1

  # double s until we have the 2**r*sth Lucas number
  while r:
    U, V = (U * V)%n, (V * V - 2 * q)%n
    q = (q * q)%n
    r -= 1

  # primality check
  # if n is prime, n divides the n+1st Lucas number, given the assumptions
  return U == 0


## Baillie-PSW ##
# this is technically a probabalistic test, but there are no known pseudoprimes
def is_bpsw(n):
  if not is_sprp(n, 2): return False

  # idea shamelessly stolen from Mathmatica's PrimeQ
  # if n is a 2-sprp and a 3-sprp, n is necessarily square-free
  if not is_sprp(n, 3): return False

  a = 5
  s = 2
  # if n is a perfect square, this will never terminate
  while legendre(a, n) != n-1:
    s = -s
    a = s-a
  return is_lucas_prp(n, a)


# an 'almost certain' primality check
def is_prime(n):
  if n < 212:
    m = binary_search(small_primes, n)
    return n == small_primes[m]

  for p in small_primes:
    if n%p == 0:
      return False

  # if n is a 32-bit integer, perform full trial division
  if n <= max_int:
    p = 211
    while p*p < n:
      for o in offsets:
        p += o
        if n%p == 0:
          return False
    return True

  return is_bpsw(n)


# next prime strictly larger than n
def next_prime(n):
  if n < 2:
    return 2

  # first odd larger than n
  n = (n + 1) | 1
  if n < 212:
    m = binary_search(small_primes, n)
    return small_primes[m]

  # find our position in the sieve rotation via binary search
  x = int(n%210)
  m = binary_search(indices, x)
  i = int(n + (indices[m] - x))

  # adjust offsets
  offs = offsets[m:] + offsets[:m]
  while True:
    for o in offs:
      if is_prime(i):
        return i
      i += o


# an infinite prime number generator
def primes(start = 0):
  for n in small_primes[start:]: yield n
  pg = primes(6)
  p = pg.next()
  q = p*p
  sieve = {221: 13, 253: 11}
  n = 211
  while True:
    for o in offsets:
      n += o
      stp = sieve.pop(n, 0)
      if stp:
        nxt = n/stp
        nxt += dindices[nxt%105]
        while nxt*stp in sieve:
          nxt += dindices[nxt%105]
        sieve[nxt*stp] = stp
      elif n < q:
        yield n
      else:
        sieve[q + dindices[p%105]*p] = p
        p = pg.next()
        q = p*p


# true if n is a prime power > 0
def is_prime_power(n):
  if n > 1:
    for p in small_primes:
      if n%p == 0:
        n /= p
        while n%p == 0: n /= p
        return n == 1

    r = isqrt(n)
    if r*r == n:
      return is_prime_power(r)

    s = icbrt(n)
    if s*s*s == n:
      return is_prime_power(s)

    p = 211
    while p*p < r:
      for o in offsets:
        p += o
        if n%p == 0:
          n /= p
          while n%p == 0: n /= p
          return n == 1

    if n <= max_int:
      while p*p < n:
        for o in offsets:
          p += o
          if n%p == 0:
            return False
      return True

    return is_bpsw(n)
  return False

Python 2 + primefac 1.1

Não tenho um Raspberry Pi para testá-lo.

from primefac import primefac

def HP(n):
    factors = list(primefac(n))

    #print n, factors

    if len(factors) == 1 and n in factors:
        return n

    n = ""
    for f in sorted(factors):
        n += str(f)
    return HP(int(n))

Experimente online

A primefacfunção retorna uma lista de todos os fatores primos de n. Em sua definição, ele chama isprime(n), que usa uma combinação de divisão de teste, método de Euler e teste de primalidade de Miller-Rabin. Eu recomendo baixar o pacote e visualizar a fonte.

Tentei usar em n = n * 10 ** int(floor(log10(f))+1) + fvez da concatenação de cadeias, mas é muito mais lento.

C #

using System;
using System.Linq;

public class Program
{
    public static void Main(string[] args) {

        Console.Write("Enter Number: ");

        Int64 n = Convert.ToInt64(Console.ReadLine());

        Console.WriteLine("Start Time: " + DateTime.Now.ToString("HH:mm:ss.ffffff"));
        Console.WriteLine("Number, Factors");

        HomePrime(n);

        Console.WriteLine("End Time: " + DateTime.Now.ToString("HH:mm:ss.ffffff"));
        Console.ReadLine();
    }

    public static void HomePrime(Int64 num) {
        string s = FindFactors(num);
        if (CheckPrime(num,s) == true) {
            Console.WriteLine("{0} is prime", num);
        } else {
            Console.WriteLine("{0}, {1}", num, s);
            HomePrime(Convert.ToInt64(RemSp(s)));
        }
    }

    public static string FindFactors(Int64 num) {
        Int64 n, r, t = num;
        string f = "";
        for (n = num; n >= 2; n--) {
            r = CalcP(n, t);
            if (r != 0) {
                f = f + " " + r.ToString();
                t = n / r;
                n = n / r + 1;
            }
        }
        return f;
    }

    public static Int64 CalcP(Int64 num, Int64 tot) {
        for (Int64 i = 2; i <= tot; i++) {
            if (num % i == 0) {
                return i;
            } 
        }
        return 0;
    }

    public static string RemSp(string str) {
        return new string(str.ToCharArray().Where(c => !Char.IsWhiteSpace(c)).ToArray());
    }

    public static bool CheckPrime(Int64 num, string s) {
        if (s == "") {
            return false;
        } else if (num == Convert.ToInt64(RemSp(s))) {
            return true;
        }
        return false;
    }

}

Esta é uma versão mais otimizada do código anterior, com algumas partes redundantes desnecessárias removidas.

Saída (no meu laptop i7):

Enter Number: 16
Start Time: 18:09:51.636445
Number, Factors
16,  2 2 2 2
2222,  2 11 101
211101,  3 11 6397
3116397,  3 163 6373
31636373 is prime
End Time: 18:09:51.637954

Teste on-line

Perl + teoria, HP (80) em 0.35s no PC

Não há Raspberry Pi disponível.

use ntheory ":all";
use feature "say";
sub hp {
  my $n = shift;
  while (!is_prime($n)) {
    $n = join "",factor($n);
  }
  $n;
}
say hp($_) for (16,20,64,65,80);

O teste de primalidade é o ES BPSW, mais uma única RM de base aleatória para números maiores. Nesse tamanho, poderíamos usar is_provable_prime(n-1 e / ou ECPP) sem diferença perceptível na velocidade, mas isso mudaria para valores de mais de 300 dígitos sem nenhum benefício real. O fatoramento inclui avaliação, potência, Rho-Brent, P-1, SQUFOF, ECM, QS, dependendo do tamanho.

Para essas entradas, ele roda aproximadamente a mesma velocidade que o código Pari / GP de Charles no site da OEIS. ntheory tem fatoração mais rápida para números pequenos, e meu P-1 e ECM são muito bons, mas o QS não é ótimo, então eu esperaria que Pari fosse mais rápido em algum momento.