I've tried playing around with Euclid's formula ($A=m^2-n^2$, $B=2mn$, $C=m^2+n^2$) but I can't see any pattern in the triples it generates or how to predict what numbers will work other than being mutually prime. Here are examples: $f(3,2)=(5,12,13), $ $f(4,1)=(15,8,17), $ $f(4,3)=(7,24,25), $ $f(5,2)=(21,20,29), $ $f(5,3)=(16,30,34), $ $f(5,4)=(9,40,41), $ $f(6,5)=(11,60,61), $
I've heard that the formula generates all primitives but I can't even figure out how to get $(3,4,5)$. How should I be using it to get, for example, the series $$(3,4,5), (5,12,13), (13,84,85), (85,3612,3613), (3613,6526884,6526885)$$ that is a natural pairing of side $C$ of one triple with side $A$ of the triple that follows? I can do it easily using other formulas but this is the $standard$ that everyone accepts.
Update: My problems in understanding are the seeming lack a pattern of the triples produced and the seeming invalidity of the statement: "Primitives will be produced if and only if $m$ and $n$ are co-prime." One comment says f(2,1)=(3,4,5) but $1$ is not prime. Let's accept it anyway, but then we find that f(3,1)=(8,6,10) is not primitive. We also find that f(5,3)=(16,30,34), f(7,3)=(40,42,58), and f(7,5)=(24,70,74) are each twice a primitive with the positions of $A$ and $B$ switched. I get a poor reception whenever I touch on the other functions I mentioned so I'm trying to find out how Euclid's formula can give me the power of prediction I seek.
Can I know the nature of the GCD(A,B,C) for a specific combination of $m$ and $n$? Can I know the difference between subsequent values of $A$ for incremental values of $m$ or $n$? Can I know the difference between $B$ and $C$ for a specific combination of $m$ and $n$? Can I find triples with matching sides, areas, or perimeters? If I can find answers to at least some of these questions, it will allow me to stop further exploration of them for a paper I'm writing. If Euclid's formula provides no such power, may I then assume that my contribution is original and perhaps non-trivial?