gmp_gcdext

(PHP 4 >= 4.0.4, PHP 5, PHP 7, PHP 8)

gmp_gcdext — Calculate GCD and multipliers

说明

gmp_gcdext(GMP|int|string $num1, GMP|int|string $num2): array

Calculates g, s, and t, such that a*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

num1: GMP 对象、int 或 string，可以按照跟在 gmp_init() 中使用字符串并自动检测 base（即当 base 等于 0 时）相同的逻辑将其解释为数字。
num2: GMP 对象、int 或 string，可以按照跟在 gmp_init() 中使用字符串并自动检测 base（即当 base 等于 0 时）相同的逻辑将其解释为数字。

返回值

An array of GMP numbers.

示例

示例 #1 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
?>

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FatPhil ¶

21 years ago

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.