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  • gmp_gcdext()

    (PHP 4 >= 4.0.4, PHP 5, PHP 7)

    Calculate GCD and multipliers

    说明

    gmp_gcdext(GMP$a, GMP$b): array

    Calculates g, s, and t, such thata*s + b*t = g = gcd(a,b), where gcd is the greatest common divisor. Returns an array with respective elements g, s and t.

    This function can be used to solve linear Diophantine equations in two variables. These are equations that allow only integer solutions and have the form:a*x + b*y = c. For more information, go to the »"Diophantine Equation" page at MathWorld

    参数

    $a

    可以是一个 GMP数据resouce,或一个可以转换为数值的字符串。

    $b

    可以是一个 GMP数据resouce,或一个可以转换为数值的字符串。

    返回值

    An array of GMP numbers.

    范例

    Solving a linear Diophantine equation

    <?php
    // Solve the equation a*s + b*t = g
    // where a = 12, b = 21, g = gcd(12, 21) = 3
    $a = gmp_init(12);
    $b = gmp_init(21);
    $g = gmp_gcd($a, $b);
    $r = gmp_gcdext($a, $b);
    $check_gcd = (gmp_strval($g) == gmp_strval($r['g']));
    $eq_res = gmp_add(gmp_mul($a, $r['s']), gmp_mul($b, $r['t']));
    $check_res = (gmp_strval($g) == gmp_strval($eq_res));
    if ($check_gcd && $check_res) {
        $fmt = "Solution: %d*%d + %d*%d = %d\n";
        printf($fmt, gmp_strval($a), gmp_strval($r['s']), gmp_strval($b),
        gmp_strval($r['t']), gmp_strval($r['g']));
    } else {
        echo "Error while solving the equation\n";
    }
    // output: Solution: 12*2 + 21*-1 = 3
    ?>
    
    The extended GCD can be used to calculate mutual modular inverses of two
    coprime numbers. Internally gmp_invert uses this extended GCD routine, 
    but effectively throws away one of the inverses.
    If gcd(a,b)=1, then r.a+s.b=1
    Therefore r.a == 1 (mod s) and s.b == 1 (mod r)
    Note that one of r and s will be negative, and so you'll want to
    canonicalise it.

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