I know that all integers are part of at least one primitive triple. But can this statement be refined to exactly one? From looking at some lists of triples it seems to be true, but I have no clue where I'd start in showing it.
Update: I'm not sure what the etiquette is regarding actually changing the question altogether… hopefully this is alright. I realised I'm actually interested only in whether any numbers act as the smallest number in more than primitive triplet.
Best Answer
Yes, a number can appear as the smallest value in two distinct primitive triples. For example, $(57, 176, 185)$ and $(57, 1624, 1625)$.
In fact, choose any two relatively prime positive integers $p$ and $q$ with $q+1 < p < q(1+\sqrt{2})$ and having opposite parity. Then $p^2-q^2 < 2pq$, and $(a,b,c)=(p^2-q^2, 2pq, p^2+q^2)$ is a primitive triple. Another primitive triple with $a$ as smallest element can be derived from the generators $r=\frac{a+1}{2}, s=\frac{a-1}{2}$: $$(r^2-s^2, 2rs, r^2+s^2) = \left(a,\frac{a^2-1}{2}, \frac{a^2+1}{2}\right).$$