code: def test_DN():
      # Most of the test cases were adapted from,
      # Solving the generalized Pell equation x**2 - D*y**2 = N, John P. Robertson, July 31, 2004.
      # https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
      # others are verified using Wolfram Alpha.
  
      # Covers cases where D <= 0 or D > 0 and D is a square or N = 0
      # Solutions are straightforward in these cases.
      assert diop_DN(3, 0) == [(0, 0)]
      assert diop_DN(-17, -5) == []
      assert diop_DN(-19, 23) == [(2, 1)]
      assert diop_DN(-13, 17) == [(2, 1)]
      assert diop_DN(-15, 13) == []
      assert diop_DN(0, 5) == []
      assert diop_DN(0, 9) == [(3, t)]
      assert diop_DN(9, 0) == [(3 * t, t)]
      assert diop_DN(16, 24) == []
      assert diop_DN(9, 180) == [(18, 4)]
      assert diop_DN(9, -180) == [(12, 6)]
      assert diop_DN(7, 0) == [(0, 0)]
  
      # When equation is x**2 + y**2 = N
      # Solutions are interchangeable
      assert diop_DN(-1, 5) == [(2, 1), (1, 2)]
      assert diop_DN(-1, 169) == [(12, 5), (5, 12), (13, 0), (0, 13)]
  
      # D > 0 and D is not a square
  
      # N = 1
      assert diop_DN(13, 1) == [(649, 180)]
      assert diop_DN(980, 1) == [(51841, 1656)]
      assert diop_DN(981, 1) == [(158070671986249, 5046808151700)]
      assert diop_DN(986, 1) == [(49299, 1570)]
      assert diop_DN(991, 1) == [
          (
              379516400906811930638014896080,
              12055735790331359447442538767,
          )
      ]
      assert diop_DN(17, 1) == [(33, 8)]
      assert diop_DN(19, 1) == [(170, 39)]
  
      # N = -1
      assert diop_DN(13, -1) == [(18, 5)]
      assert diop_DN(991, -1) == []
      assert diop_DN(41, -1) == [(32, 5)]
      assert diop_DN(290, -1) == [(17, 1)]
      assert diop_DN(21257, -1) == [(13913102721304, 95427381109)]
      assert diop_DN(32, -1) == []
  
      # |N| > 1
      # Some tests were created using calculator at
      # http://www.numbertheory.org/php/patz.html
  
      assert diop_DN(13, -4) == [(3, 1), (393, 109), (36, 10)]
      # Source I referred returned (3, 1), (393, 109) and (-3, 1) as fundamental solutions
      # So (-3, 1) and (393, 109) should be in the same equivalent class
      assert equivalent(-3, 1, 393, 109, 13, -4) == True
  
      assert diop_DN(13, 27) == [
          (220, 61),
          (40, 11),
          (768, 213),
          (12, 3),
      ]
      assert set(diop_DN(157, 12)) == {
          (13, 1),
          (10663, 851),
          (579160, 46222),
          (483790960, 38610722),
          (26277068347, 2097138361),
          (21950079635497, 1751807067011),
      }
      assert diop_DN(13, 25) == [(3245, 900)]
      assert diop_DN(192, 18) == []
      assert diop_DN(23, 13) == [(-6, 1), (6, 1)]
      assert diop_DN(167, 2) == [(13, 1)]
      assert diop_DN(167, -2) == []
  
      assert diop_DN(123, -2) == [(11, 1)]
      # One calculator returned [(11, 1), (-11, 1)] but both of these are in
      # the same equivalence class
      assert equivalent(11, 1, -11, 1, 123, -2)
  
      assert diop_DN(123, -23) == [(-10, 1), (10, 1)]
  
      assert diop_DN(0, 0, t) == [(0, t)]
      assert diop_DN(0, -1, t) == []
  
imports: from sympy.core.add import Add
  from sympy.core.mul import Mul
  from sympy.core.numbers import Rational, oo, pi
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import symbols
  from sympy.matrices.dense import Matrix
  from sympy.ntheory.factor_ import factorint
  from sympy.simplify.powsimp import powsimp
  from sympy.core.function import _mexpand
  from sympy.core.sorting import default_sort_key, ordered
  from sympy.functions.elementary.trigonometric import sin
  from sympy.solvers.diophantine import diophantine
  from sympy.solvers.diophantine.diophantine import diop_DN, diop_solve, diop_ternary_quadratic_normal, diop_general_pythagorean, diop_ternary_quadratic, diop_linear, diop_quadratic, diop_general_sum_of_squares, diop_general_sum_of_even_powers, descent, diop_bf_DN, divisible, equivalent, find_DN, ldescent, length, reconstruct, partition, power_representation, prime_as_sum_of_two_squares, square_factor, sum_of_four_squares, sum_of_three_squares, transformation_to_DN, transformation_to_normal, classify_diop, base_solution_linear, cornacchia, sqf_normal, gaussian_reduce, holzer, check_param, parametrize_ternary_quadratic, sum_of_powers, sum_of_squares, _diop_ternary_quadratic_normal, _nint_or_floor, _odd, _even, _remove_gcd, _can_do_sum_of_squares, DiophantineSolutionSet, GeneralPythagorean, BinaryQuadratic
  from sympy.testing.pytest import slow, raises, XFAIL
  from sympy.utilities.iterables import signed_permutations
label_desc: Use image node_id 15 for calling a global function with example usage: test_DN() without return types