JavaScript Fun Question: Diophantine Equation

In mathematics, the Diophantine equation is a polynomial equation that usually has two or more unknowns and requires their integer solutions.

Given the following Diophantine equation, find all its positive integer solutions.

x² - 4y² = n

x and y are unknowns, and n is a given constant. The solution set of x, y will be displayed using the following nested array:

[[x1, y1], [x2, y2] ....]

Here are some examples:

sol_equa(90005) --> [[45003, 22501], [9003, 4499], [981, 467], [309, 37]]

sol_equa(90002) --> []

Let’s see how to solve this problem. First look at the left side of this equation, x² - 4y². At first glance, you will have a feeling that it can be transformed is (x - 2y) * (x + 2y), when you think of this, you have taken the first step.

Because the constant N on the right side of the equation may be a very large number, if the exhaustive method is used, the efficiency is very low.

We can try to decompose this constant and factor it into two terms.

For example, N=24, it can be decomposed into two items as follows:

[1,24], [2,12], [3,8], [4,6 ]

We use these possibilities to apply it to the formula:

x - 2y = 1

x + 2y = 24

----- ---------

x - 2y = 2

x + 2y = 12

......

This way It is transformed into finding a linear equation of two variables.

Finally, we can select the positive integer solution.

function solequa(n) {  
    var result = [];  
    for(var a=1,b=n;a<=b;a++){  
        if(n % a == 0){  
            b = n / a;  
            var x = (a + b) / 2;  
            var y = (b - a) / 4;  
            if(parseInt(x) == x && parseInt(y) == y && x >=0 && y >= 0){  
                result.push([x,y]);  
            }  
        }  
    }  
    return result;  
}

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