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

The bounds of x and y are an important part of the problem. I personally went with this Wolfram Alpha query and checked the exact forms of the variables.

Thanks to @Bleep-Bloop and comments, a very elegant bound optimization was found, which is x < n and x <= y < n - x. The results are the same and the times are nearly identical.

Also, since the only possible values for x and y are positive even integers, we can reduce the amount of loop iterations by half.

To optimize even further, since we compute the upper bound of x, we build a list of all possible values for x and make the computation parallel. That saves a massive amount of time on higher values of N but it's a bit slower for smaller values because of the overhead of the parallelization.

Here's the final code:

Non-parallel version, with int values:

List<string> res = new List<string>();
int n2 = n * n;

double maxX = 0.5 * (2.0 * n - Math.Sqrt(2) * Math.Sqrt(n2 + 1));

for (int x = 2; x < maxX; x += 2)
{
    int maxY = (int)Math.Floor((n2 - 2.0 * n * x - 1.0) / (2.0 * n - 2.0 * x));

    for (int y = x; y <= maxY; y += 2)
    {
        int z2 = x * x + y * y + 1;
        int z = (int)Math.Sqrt(z2);

        if (z * z == z2 && x + y + z <= n)
            res.Add(x + "," + y + "," + z);
    }
}

Parallel version, with long values:

using System.Linq;

...

// Use ConcurrentBag for thread safety
ConcurrentBag<string> res = new ConcurrentBag<string>();
long n2 = n * n;

double maxX = 0.5 * (2.0 * n - Math.Sqrt(2) * Math.Sqrt(n2 + 1L));

// Build list to parallelize
int nbX = Convert.ToInt32(maxX);
List<int> xList = new List<int>();
for (int x = 2; x < maxX; x += 2)
    xList.Add(x);

Parallel.ForEach(xList, x =>
{
    int maxY = (int)Math.Floor((n2 - 2.0 * n * x - 1.0) / (2.0 * n - 2.0 * x));

    for (long y = x; y <= maxY; y += 2)
    {
        long z2 = x * x + y * y + 1L;
        long z = (long)Math.Sqrt(z2);

        if (z * z == z2 && x + y + z <= n)
            res.Add(x + "," + y + "," + z);
    }
});

When ran individually on a i5-8400 CPU, I get these results:

N: 10; Solutions: 1; Time elapsed: 0.03 ms (Not parallel, int)

N: 100; Solutions: 6; Time elapsed: 0.05 ms (Not parallel, int)

N: 1000; Solutions: 55; Time elapsed: 0.3 ms (Not parallel, int)

N: 10000; Solutions: 543; Time elapsed: 13.1 ms (Not parallel, int)

N: 100000; Solutions: 5512; Time elapsed: 849.4 ms (Parallel, long)

---

You must use long when N is greater than 36340, because when it's squared, it overflows an int's max value. Finally, the parallel version starts to get better than the simple one when N is around 23000, with ints.

Answer 2

#include<iostream>
#include<math.h>
int main()
{
    int N = 10000;
    int c = 0;
    for (int x = 2; x < N; x+=2)
    {
        for (int y = x; y < (N - x); y+=2)
        {
            auto z = sqrt(x * x + y * y + 1);
            if(x+y+z>N){
                break;
            }
            if (z - (int) z == 0)
            {
                c++;
            }
        }
    }
    std::cout<<c;
}

This is my solution. On testing the previous solutions for this problem I found that x,y are always even and z is odd. I dont know the mathematical nature behind this, I am currently trying to figure that out.

Answer 3

No time to properly test it, but seemed to yield the same results as your code (at 100 -> 6 results and at 1000 -> 55 results).

With N=1000 a time of 2ms vs your 144ms also without List

and N=10000 a time of 28ms

var N = 1000;
var c = 0;

for (int x = 2; x < N; x+=2)
{
    for (int y = x; y < (N - x); y+=2)
    {
        long z2 = x * x + y * y + 1;
        int z = (int) Math.Sqrt(z2);
        if (x + y + z > N)
            break;
        if (z * z == z2)
            c++;
    }
}

Console.WriteLine(c);

Answer 4

Here is a simple improvement in Python (converting to the faster equivalent in C-based code is left as an exercise for the reader). To get accurate timing for the computation, I removed printing the solutions themselves (after validating them in a previous run).

• Use an outer loop for one free variable (I chose z), constrained only by its relation to N.
• Use an inner loop (I chose y) constrained by the outer loop index.
• The third variable is directly computed per requirement 2.

Timing results:

-------------------- 10 
 1 solutions found in 2.3365020751953125e-05  sec.
-------------------- 100 
 6 solutions found in 0.00040078163146972656  sec.
-------------------- 1000 
 55 solutions found in 0.030081748962402344  sec.
-------------------- 10000 
 543 solutions found in 2.2078349590301514  sec.
-------------------- 100000 
 5512 solutions found in 214.93411707878113  sec.

That's 3:35 for the large case, plus your time to collate and/or print the results.

If you need faster code (this is still pretty brute-force), look into Diophantine equations and parameterizations to generate (y, x) pairs, given the target value of z^2 - 1.

import math
import time

def break3(N):
    """
    10 <= N <= 10^5
    return x, y, z triples such that:
        x <= y <= z
        x^2 + y^2 = z^2 - 1        
        x + y + z <= N
    """

    """
    Observations:
    z <= x + y
    z < N/2
    """

    count = 0
    z_limit = N // 2
    for z in range(3, z_limit):

        # Since y >= x, there's a lower bound on y
        target = z*z - 1
        ymin = int(math.sqrt(target/2))
        for y in range(ymin, z):
            # Given y and z, compute x.
            # That's a solution iff x is integer.
            x_target = target - y*y
            x = int(math.sqrt(x_target))
            if x*x == x_target and x+y+z <= N:
                # print("solution", x, y, z)
                count += 1

    return count


test = [10, 100, 1000, 10**4, 10**5]
border = "-"*20

for case in test: 
    print(border, case)
    start = time.time()
    print(break3(case), "solutions found in", time.time() - start, "sec.")

Answer 5

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

Answer 6

I want to get it done in C# and it should be covering all the test cases based on condition provided in the question.

The basic code, converted to long to process the N <= 100000 upper limit, with every optimizaion thrown in I could. I used alternate forms from @Mat's (+1) Wolfram Alpha query to precompute as much as possible. I also did a minimal perfect square test to avoid millions of sqrt() calls at the upper limit:

public static void Main()
{
    int c = 0;

    long N = long.Parse(Console.ReadLine());
    long N_squared = N * N;

    double half_N_squared = N_squared / 2.0 - 0.5;
    double x_limit = N - Math.Sqrt(2) / 2.0 * Math.Sqrt(N_squared + 1);

    for (long x = 2; x < x_limit; x += 2)
    {
        long x_squared = x * x + 1;

        double y_limit = (half_N_squared - N * x) / (N - x);

        for (long y = x; y < y_limit; y += 2)
        {
            long z_squared = x_squared + y * y;
            int digit = (int) z_squared % 10;

            if (digit == 3 || digit == 7)
            {
                continue;  // minimalist non-perfect square elimination
            }

            long z = (long) Math.Sqrt(z_squared);

            if (z * z == z_squared)
            {
                c++;
            }
        }
    }

    Console.WriteLine(c);
}

I followed the trend and left out "the degenerate solution" as implied by the OP's code but not explicitly stated.