How to descend within the “Tree of primitive Pythagorean triples”? It is well-known that the set of all primitive Pythagorean triples has the structure of an infinite ternary rooted tree. What is the exact algorithm ( _i.e._ , formula, or possibly set of three formulas) by which one can take a given Pythagorean triple $(a,b,c)$ and find the immediately smaller triple in the tree? For example, given $(165,52,173)$, how does one obtain its [unique] “ancestor” triple $(77,36,85)$?

Starting with $(165,52,173)$, we attempt to find the $p,q$ pair which generates this triple, i.e., $p^2-q^2=165,2pq=52,p^2+q^2=173$. Clearly, we have $2q^2=173-165=8\implies q=2$ and thus $p=13$.

The ancestor of this triple arises from $(|p-2q|,q)\text{ or }(q,|p-2q|)$, whichever places $p',q'$ in largest-to-smallest order. In this case, we have $p-2q=9$ and thus the ancestor pair is $(p',q')=(9,2)$ and therefore the ancestor triple is $(p'^2-q'^2,2p'q',p'^2+q'^2)=(77,36,85)$.