Let $T$ be the set of pythagorean triples, that is, triples of integers (a,b,c) satisfying a2 + b2 = c2. We think of $T$ as the set of right angles triangles with integer lengths. And let $f : T \rightarrow \mathbb{Z}$ be the function $(a,b,c) \mapsto \frac{ab}{12}$ which computes the area of a triangle (divided by 6, which seems to always be a factor for some reason). 


I was wondering: what are the number theoretic propertires of $f$? It seems to produce numbers with few prime factors. What is the reason for this? For instance, $f(3,4,5) = 1$, $f(36,77,85) = 3 \* 11 \* 7$, and $f(65,72,97)=39\*5\*2$. Can we put a bound on the number of prime factors in the numbers that $f$ spits out? Or at least, can we give a 'generic' statement such as 'The output of $f$ almost always spits out numbers with less than 8 factors' or something?



Does the function which sends a right angled triangle to its area produce infinitely many numbers having hardly any prime factors?
Presumably, there are infinitely many primes $p$, $q$, such that $p+q$ is 6 times a prime and $p-q$ is 4 times a prime (e.g., $73+5=6\times13$, $73-5=4\times17$). This ought to follow from the prime $k$-tuples conjecture. Proving it is another matter. 


Slightly simpler (but still out of reach), there should be infinitely many primes $p$, $q$, such that $2p+q$ is prime and $2p-q$ is 3 times a prime (e.g., $p=13$, $q=5$). Then the integer right-triangle with sides $4pq$ and $4p^2-q^2$ has area 6 times a product of 4 primes.


EDIT: Maybe not prime $k$-tuples but Schinzel's Hypothesis H is what's needed. 


