What is the Equation for every possible solution to the Pythagorean Theorem where a and b are x and y coordinates on a graph.

Question

I currently am working on a minor programming project, and I have a program that can plot points in the problem above. However, this requires a lot of computing power that the average school chromebook simply cannot output. I realized that the points were very specifically placed and followed a pattern, but I have no idea where to start linking the info together. [Solutions from the range (-30,-30) to (30,30)][1] If anyone is interested in the code:

var a;
var b;
var c;
var list=[];
// Find A and B
for(var i = -30; i<=30; i++){
  a=i;
  console.log(a);
 for(var j = -30; j<=30; j++){
  b=j;
  console.log(b);
  wholenum(a, b);
  }
}


// Test If a^2+b^2 makes a whole number
function wholenum(a, b) {
  if(Math.floor(Math.sqrt(a^2+b^2)) === Math.sqrt(a^2+b^2)){
    c=Math.sqrt(a^2+b^2);
    appendItem(list, "("+a+","+b+")");
   // testpyth(a, b, c);
  }
}

I beleive it could be solutions to a system of a number of equations, but im not sure. Thank you in advance for any help. [1]: https://i.stack.imgur.com/0GIKG.png

Answer 1

There are many formulas to generate Pythagorean triples and the most common one is Euclid's, which works but requires some restrictions on input to avoid generating trivial and imprimitive triples. $$A=m^2-k^2\qquad B=2mk\qquad C=m^2+k^2$$

There is another formula that generates a Pythagorean triple for any pair of natural numbers $\space (n,k)\space$ with fewer imprimitives than any other formula I have seen.

\begin{align*} &A=(2n-1)^2+&&2(2n-1)k\\ &B= &&2(2n-1)k+2k^2\\ &C=(2n-1)^2+&&2(2n-1)k+2k^2 \end{align*}

Using either formula, we will have $$X=A\quad\text{and}\quad Y=B$$

and we can use these to generate a scatter plot similar to the one shown here.