How to enumerate x^2 + y^2 = z^2 - 1 (with additional constraints)

Question

Lets N be a number (10<=N<=10^5).

I have to break it into 3 numbers (x,y,z) such that it validates the following conditions.

Answer 1

Here's a method that enumerates the triples, rather than exhaustively testing for them, using number theory as described here: https://mathoverflow.net/questions/29644/enumerating-ways-to-decompose-an-integer-into-the-sum-of-two-squares

Since the math took me a while to comprehend and a while to implement (gathering some code that's credited above it), and since I don't feel much of an authority on the subject, I'll leave it for the reader to research. This is based on expressing numbers as Gaussian integer conjugates. (a + bi)*(a - bi) = a^2 + b^2. We first factor the number, z^2 - 1, into primes, decompose the primes into Gaussian conjugates and find different expressions that we expand and simplify to get a + bi, which can be then raised, a^2 + b^2.

A perk of reading about the Sum of Squares Function is discovering that we can rule out any candidate z^2 - 1 that contains a prime of form 4k + 3 with an odd power. Using that check alone, I was able to reduce Prune's loop on 10^5 from 214 seconds to 19 seconds (on repl.it) using the Rosetta prime factoring code below.

The implementation here is just a demonstration. It does not have handling or optimisation for limiting x and y. Rather, it just enumerates as it goes. Play with it here.

Python code:

# https://math.stackexchange.com/questions/5877/efficiently-finding-two-squares-which-sum-to-a-prime
def mods(a, n):
    if n <= 0:
        return "negative modulus"
    a = a % n
    if (2 * a > n):
        a -= n
    return a

def powmods(a, r, n):
    out = 1
    while r > 0:
        if (r % 2) == 1:
            r -= 1
            out = mods(out * a, n)
        r /= 2
        a = mods(a * a, n)
    return out

def quos(a, n):
    if n <= 0:
        return "negative modulus"
    return (a - mods(a, n))/n

def grem(w, z):
    # remainder in Gaussian integers when dividing w by z
    (w0, w1) = w
    (z0, z1) = z
    n = z0 * z0 + z1 * z1
    if n == 0:
        return "division by zero"
    u0 = quos(w0 * z0 + w1 * z1, n)
    u1 = quos(w1 * z0 - w0 * z1, n)
    return(w0 - z0 * u0 + z1 * u1,
           w1 - z0 * u1 - z1 * u0)

def ggcd(w, z):
    while z != (0,0):
        w, z = z, grem(w, z)
    return w

def root4(p):
    # 4th root of 1 modulo p
    if p <= 1:
        return "too small"
    if (p % 4) != 1:
        return "not congruent to 1"
    k = p/4
    j = 2
    while True:
        a = powmods(j, k, p)
        b = mods(a * a, p)
        if b == -1:
            return a
        if b != 1:
            return "not prime"
        j += 1

def sq2(p):
    if p % 4 != 1:
      return "not congruent to 1 modulo 4"
    a = root4(p)
    return ggcd((p,0),(a,1))

# https://rosettacode.org/wiki/Prime_decomposition#Python:_Using_floating_point
from math import floor, sqrt

def fac(n):
    step = lambda x: 1 + (x<<2) - ((x>>1)<<1)
    maxq = long(floor(sqrt(n)))
    d = 1
    q = n % 2 == 0 and 2 or 3 
    while q <= maxq and n % q != 0:
        q = step(d)
        d += 1
    return q <= maxq and [q] + fac(n//q) or [n]

# My code...
# An answer for  https://stackoverflow.com/questions/54110614/

from collections import Counter
from itertools import product
from sympy import I, expand, Add

def valid(ps):
  for (p, e) in ps.items():
    if (p % 4 == 3) and (e & 1):
      return False
  return True

def get_sq2(p, e):
  if p == 2:
    if e & 1:
      return [2**(e / 2), 2**(e / 2)]
    else:
      return [2**(e / 2), 0]
  elif p % 4 == 3:
    return [p, 0]
  else:
    a,b = sq2(p)
    return [abs(a), abs(b)]

def get_terms(cs, e):
  if e == 1:
    return [Add(cs[0], cs[1] * I)]
  res = [Add(cs[0], cs[1] * I)**e]
  for t in xrange(1, e / 2 + 1):
    res.append(
      Add(cs[0] + cs[1]*I)**(e-t) * Add(cs[0] - cs[1]*I)**t)
  return res

def get_lists(ps):
  items = ps.items()
  lists = []
  for (p, e) in items:
    if p == 2:
      a,b = get_sq2(2, e)
      lists.append([Add(a, b*I)])
    elif p % 4 == 3:
      a,b = get_sq2(p, e)
      lists.append([Add(a, b*I)**(e / 2)])
    else:
      lists.append(get_terms(get_sq2(p, e), e))
  return lists


def f(n):
  for z in xrange(2, n / 2):
    zz = (z + 1) * (z - 1)
    ps = Counter(fac(zz))
    is_valid = valid(ps)
    if is_valid:
      print "valid (does not contain a prime of form\n4k + 3 with an odd power)"
      print "z: %s, primes: %s" % (z, dict(ps))
      lists = get_lists(ps)
      cartesian = product(*lists)
      for element in cartesian:
        print "prime square decomposition: %s" % list(element)
        p = 1
        for item in element:
          p *= item
        print "complex conjugates: %s" % p
        vals = p.expand(complex=True, evaluate=True).as_coefficients_dict().values()
        x, y = vals[0], vals[1] if len(vals) > 1 else 0
        print "x, y, z: %s, %s, %s" % (x, y, z)
        print "x^2 + y^2, z^2-1: %s, %s" % (x**2 + y**2, z**2 - 1)
      print ''

if __name__ == "__main__":
  print f(100)

Output:

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 3, primes: {2: 3}
prime square decomposition: [2 + 2*I]
complex conjugates: 2 + 2*I
x, y, z: 2, 2, 3
x^2 + y^2, z^2-1: 8, 8

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 9, primes: {2: 4, 5: 1}
prime square decomposition: [4, 2 + I]
complex conjugates: 8 + 4*I
x, y, z: 8, 4, 9
x^2 + y^2, z^2-1: 80, 80

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 17, primes: {2: 5, 3: 2}
prime square decomposition: [4 + 4*I, 3]
complex conjugates: 12 + 12*I
x, y, z: 12, 12, 17
x^2 + y^2, z^2-1: 288, 288

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 19, primes: {2: 3, 3: 2, 5: 1}
prime square decomposition: [2 + 2*I, 3, 2 + I]
complex conjugates: (2 + I)*(6 + 6*I)
x, y, z: 6, 18, 19
x^2 + y^2, z^2-1: 360, 360

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 33, primes: {17: 1, 2: 6}
prime square decomposition: [4 + I, 8]
complex conjugates: 32 + 8*I
x, y, z: 32, 8, 33
x^2 + y^2, z^2-1: 1088, 1088

valid (does not contain a prime of form
4k + 3 with an odd power)
z: 35, primes: {17: 1, 2: 3, 3: 2}
prime square decomposition: [4 + I, 2 + 2*I, 3]
complex conjugates: 3*(2 + 2*I)*(4 + I)
x, y, z: 18, 30, 35
x^2 + y^2, z^2-1: 1224, 1224