How to sensibly use Euclid's formula for Pythagorean triples.

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?

• You may find of interest the very beautiful reflective generation of the tree of primitive Pythagorean tripes - which is a nice simple example of some beautiful deeper connections between number theory and geometry. Follow the link to learn more. – Bill Dubuque Apr 19 at 17:11
• @Somos Thanks for the answer but the link doesn't work in my browser. It says: This site can’t be reached – poetasis Apr 19 at 17:21