丢番图方程求解器

简介

本程序是一个丢番图方程（Diophantine Equation）求解器，可以解决形如Ax + By = C的整数方程的解。

本程序使用的算法是扩展欧几里得算法（Extended Euclidean Algorithm），通过求得Ax + By = gcd(A,B)的一组特解，再由此推导出Ax + By = C的一组特解。进而通过枚举x和y来求得所有解。

使用方法

本程序接收用户输入的A、B、C三个参数，并输出方程的所有解。若方程无解，则输出"No solution."

def diophantine_equation_solver(A, B, C):
    """
    Solve the Diophantine Equation Ax + By = C.
    * A: int, the coefficient of x.
    * B: int, the coefficient of y.
    * C: int, the constant on the right-hand side of equation.
    * return: list of tuples, containing all solutions in the form of (x, y).
    """
    # check if the equation has solution
    gcd, x, y = extended_euclidean_algorithm(A, B)
    if C % gcd != 0:
        return "No solution."
    else:
        # get a particular solution of Ax + By = gcd(A,B)
        x0 = x * (C // gcd)
        y0 = y * (C // gcd)
        
        # generate all solutions
        solutions = []
        k = 0
        while x0 + k * (B // gcd) >= 0:
            # solutions are in the form of (x, y) 
            solutions.append((x0 + k * (B // gcd), y0 - k * (A // gcd)))
            k -= 1
        k = 1
        while x0 + k * (B // gcd) <= 0:
            solutions.append((x0 + k * (B // gcd), y0 - k * (A // gcd)))
            k += 1
        return solutions        

程序实现

扩展欧几里得算法

def extended_euclidean_algorithm(a, b):
    """
    Calculate the gcd(a, b) and a solution (x,y)
    to the equation ax + by = gcd(a,b).
    * a: int, first positive integer.
    * b: int, second positive integer.
    * return: tuple, (gcd(a,b), x, y).
    """
    if a == 0:
        return (b, 0, 1)
    else:
        gcd, y, x = extended_euclidean_algorithm(b % a, a)
        return (gcd, x - (b // a) * y, y)

丢番图方程解法

def diophantine_equation_solver(A, B, C):
    """
    Solve the Diophantine Equation Ax + By = C.
    * A: int, the coefficient of x.
    * B: int, the coefficient of y.
    * C: int, the constant on the right-hand side of equation.
    * return: list of tuples, containing all solutions in the form of (x, y).
    """
    # check if the equation has solution
    gcd, x, y = extended_euclidean_algorithm(A, B)
    if C % gcd != 0:
        return "No solution."
    else:
        # get a particular solution of Ax + By = gcd(A,B)
        x0 = x * (C // gcd)
        y0 = y * (C // gcd)
        
        # generate all solutions
        solutions = []
        k = 0
        while x0 + k * (B // gcd) >= 0:
            # solutions are in the form of (x, y) 
            solutions.append((x0 + k * (B // gcd), y0 - k * (A // gcd)))
            k -= 1
        k = 1
        while x0 + k * (B // gcd) <= 0:
            solutions.append((x0 + k * (B // gcd), y0 - k * (A // gcd)))
            k += 1
        return solutions   

示例

>>> diophantine_equation_solver(10, 6, 14)
[(7, 0), (1, 2), (-5, 4), (-11, 6), (-17, 8), (-23, 10), (-29, 12), (-35, 14)]

>>> diophantine_equation_solver(13, 7, 19)
"No solution."

总结

本程序通过实现扩展欧几里得算法和枚举解的方法，能够高效地求解丢番图方程。其时间复杂度为O(log(min(A,B)))，在实际应用中具有较高的可用性。