# The Mordell-Schinzel conjecture for cubic surfaces
# János Kollár, Jennifer Li
# Last updated: Sunday, November 24, 2024
# This program, integerTriples.py, computes the integer solutions of the four equations (11.1-11.4) of the paper.
# The output consists of triples (x, y, z) where x, y, z are integers.

# Here, we define arrays that will store the integer solutions as we find them.
eq_1_integer_solutions = []
eq_2_integer_solutions = []
eq_3_integer_solutions = []
eq_4_integer_solutions = []

# We define the four equations (11.1-11.4) here.
def eq_1(x, y):
    if x == 0 or y == 0:
        return False

    return (x**3 + y**3 + 1 - x**2 - y**2) / (x * y)

def eq_2(x, y):
    # z = (x**3 + y**3 + 1 -2*x**2 - x - 2*y**2 - y) / (x * y)
    if x == 0 or y == 0:
        return False

    return (x**3 + y**3 + 1 - 2*x**2 - x - 2*y**2 - y) / (x * y)


def eq_3(x, y):
    # z = (x**3 + y**3 + 1 -2x**2 - x - y**2) / (x * y)
    if x == 0 or y == 0:
        return False
    
    return (x**3 + y**3 + 1 - 2*x**2 - x - y**2) / (x * y)

def eq_4(x, y):
    # z = (x**3 + y**3 + 1 -x**2 - 2*y**2 - y) / (x * y)
    if x == 0 or y == 0:
        return False

    return (x**3 + y**3 + 1 -x**2 - 2*y**2 - y) / (x * y)

# This ensures that all integers show up as integers without a decimal (by default, Python will display the integer 1 as 1.0, for example).
def show_as_int(set):
    return tuple(int(num) for num in set)

# This prints each solution on a separate line.
def print_result(list):
    for r in list:
        print(f"   * {show_as_int(r)}")

# Here we find the integer solutions for the equations (11.1-11.4).
# Instead of taking the range |x|, |y| < 100, we take the range |x|, |y| < 1000 (it only takes a little longer).
# Each time we find an integer solution, we add it to the appropriate array (there is one array for each of the four equations, defined above).
for x in range(-1000, 1000):
    for y in range(-1000, 1000):
        z = eq_1(x, y)
        if z.is_integer() and z != False:
            eq_1_integer_solutions.append((x,y,z))

        z = eq_2(x, y)
        if z.is_integer() and z != False:
            eq_2_integer_solutions.append((x,y,z))

        z = eq_3(x, y)
        if z.is_integer() and z != False:
            eq_3_integer_solutions.append((x,y,z))

        z = eq_4(x, y)
        if z.is_integer() and z != False:
            eq_4_integer_solutions.append((x,y,z))

# The remaining lines print the integer triples (x, y, z) for each equation.
print("Integer triples (x, y, z) for the first equation (11.1):")
print_result(eq_1_integer_solutions)

print("Integer triples (x, y, z) for the second equation (11.2):")
print_result(eq_2_integer_solutions)

print("Integer triples (x, y, z) for the third equation (11.3):")
print_result(eq_3_integer_solutions)

print("Integer triples (x, y, z) for the fourth equation (11.4):")
print_result(eq_4_integer_solutions)