TITLE: Prime Number in triangle
QUESTION [5 upvotes]: I had a question here, the measures of the sides of a right triangle (a single unit) can be prime numbers? If they can not, why?! But, if you can, could you help me find an example?

REPLY [2 votes]: Note that primitive pythagorean triples (ones where the three sides have no common factor, so the only hope of finding primes - given that there is no primitive triple with side $1$ and all sides non-zero) are obtained by taking coprime integers $p\gt q\ge1 $, one even, and one odd.
The sides of the triangle are then $p^2+q^2$, $2pq$ and $p^2-q^2$.
We have $2pq$ divisible by $4$.
If either $p$ or $q$ is divisible by $3$ then $2pq$ is divisible by $3$, otherwise $p^2-q^2$ is divisible by $3$.
The squares modulo $5$ are $0, \pm 1$. If $5|pq$ then $5|2pq$. If not then (mod $5$) we have one $p^2 \pm q^2\equiv 0$. So one side is divisible by $5$.
So, as is sometimes said, the product of the sides is divisible by $60$. If two sides are prime then we have the examples $3$, $4$, $5$ and $5$, $12$, $13$. Other than these, to get two primes we need $60|2pq$ as in $11$, $60$, $61$.