Is a program that finds all pythagorean triples Involving one number useful?

Question

I created a program that quickly finds all the pythagorean triples for one number (up to about 14 digits currently). Is this useful or is it fairly obvious to do? I'm a lay person in math but love patterns, numbers and research. If it's useful I will make it available.
Example: all of the pythagorean triples for:

29173 (prime):
$29173^2 = 27948^2 + 8365^2$
$29173^2 + 425531964^2 = 425531965^2$

Or the number 221:
$221^2 = 204^2 + 85^2$
$221^2 = 195^2 + 104^2$
$221^2 = 171^2 + 140^2$
$221^2 + 60^2 = 229^2$
$221^2 + 1428^2 = 1445^2$
$221^2 + 1872^2 = 1885^2$
$221^2 + 24420^2 = 24421^2$

Answer 1

The “interesting” triples are the "primitives" and there are many ways to find these, even among the related links to the right of this page. All others can be found using the factors of "given" numbers.

Things to keep in mind are:

1. Side A can be any odd number greater than one.
2. Side B can be any multiple of four.
3. Side C "must" be of the form $\,4n+1\,$ but not all such numbers qualify for primitives. Here is a list of the first couple of dozen primitive hypotenuse values.

If you like patterns, note that primitives (and a few imprimitives) can be grouped into sets with at least 2 common attributes in each row and column of the sample table below. If you study the patterns, you may see how the formula that generates them was developed.

\begin{array}{c|c|c|c|c|c|c|} n & k=1 & k=2 & k=3 & k=4 & k=5 \\ \hline Set_1 & 3,4,5 & 5,12,13& 7,24,25& 9,40,41& 11,60,61 \\ \hline Set_2 & 15,8,17 & 21,20,29 &27,36,45 &33,56,65 & 39,80,89 \\ \hline Set_3 & 35,12,37 & 45,28,53 &55,48,73 &65,72,97 & 75,100,125 \\ \hline Set_{4} &63,16,65 &77,36,85 &91,60,109 &105,88,137 &119,120,169 \\ \hline Set_{5} &99,20,101 &117,44,125 &135,72,153 &153,104,185 &171,140,221 \\ \hline \end{array}

If Euclid's formula is considered functions of $(m,k)$, then these were generated with the same functions using $\big((2n-1+k),k\big)$ which resolve to.

\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*}