Pythagorean Triples

Three integers $x$, $y$, and $z$ are Pythagorean triples if they satisfy

\[\begin{aligned} x^2+y^2&=z^2 \end{aligned}\]

To avoid duplicates, we assume $x<y$.

QUBO++ program for listing Pythagorean Triples

The following program lists Pythagorean triples with $x\leq 16$, $y\leq 16$, and $z\leq 16$:

#include "qbpp.hpp"
#include "qbpp_easy_solver.hpp"

int main() {
  auto x = 1 <= qbpp::var_int("x") <= 16;
  auto y = 1 <= qbpp::var_int("y") <= 16;
  auto z = 1 <= qbpp::var_int("z") <= 16;
  auto f = x * x + y * y - z * z == 0;
  auto c = 1 <= y - x <= +qbpp::inf;
  auto g = f + c;
  g.simplify_as_binary();
  auto solver = qbpp::easy_solver::EasySolver(g);
  solver.time_limit(10.0);
  solver.enable_best_energy_sols(10);
  auto sol = solver.search();
  for (const auto& best_sol : sol.sols()) {
    std::cout << "x=" << best_sol(x) << ", y=" << best_sol(y)
              << ", z=" << best_sol(z) << ", *f=" << best_sol(*f)
              << ", *c=" << best_sol(*c) << std::endl;
  }
}

In this program, we define integer variables x, y, and z with ranges from 1 to 16. We then create two constraint expressions:

f for $x^2+y^2-z^2=0$, and
c for $x+1\leq y$.

They are combined into g. The expression g attains its minimum value 0 when all constraints are satisfied.

An Easy Solver object solver is created for g and configured with the following options:

time_limit(10.0): Terminates the search after 10 seconds.
enable_best_energy_sols(10): Keeps up to 10 solutions with the best (lowest) energy.

The call to search() returns a result object sol. The member function call sol.sols() provides access to the stored best-energy solutions, which are printed using a range-based for loop.

This program produces output like the following:

x=3, y=4, z=5, *f=0, *c=1
x=6, y=8, z=10, *f=0, *c=2
x=9, y=12, z=15, *f=0, *c=3
x=5, y=12, z=13, *f=0, *c=7