How many primitive Pythagorean triples are found in the interior of Pascal's triangle?
By "interior", I mean the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. I do not require that a triple is found in a single row of the triangle.
There are certainly nonprimitive Pythagorean triples in the interior. For example, $\left(\binom{7}{2},\binom{8}{2},\binom{7}{3}\right)$ and $\left(\binom{13}{5},\binom{13}{6},\binom{66}{2}\right)$, which both simplify to $(3,4,5)$. Or $\left(\binom{7}{3},\binom{9}{3},\binom{14}{2}\right)$, which simplifies to $(5,12,13)$.
I used Excel to find that none of the first $1325$ primitive Pythagorean triples (in nondecreasing order of perimeter) are found in the interior.
Of those $1325$ primitive Pythagorean triples:
- $1101$ of them have no numbers in the interior (for example, for the triple $(11,60,61)$, none of those numbers is in the interior)
- $216$ of them have exactly one number in the interior (for example, for the triple $(8,15,17)$ only $15$ is in the interior)
- $8$ of them have exactly two numbers in the interior (for example, for the triple $(20,21,29)$ only $20$ and $21$ are in the interior)
Possibly related: Here is a proof that there is exactly one Pythagorean triple (not necessarily primitive) consisting of consecutive terms in a row of Pascal's triangle.
Context: This is a follow-up question on my earlier question, "Does any row of Pascal's triangle contain a Pythagorean triple?" (not necessarily primitive)
