Prime Number in triangle

Question

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?

Answer 1

They cannot be all prime. For suppose that $x^2+y^2=z^2$, where $x$, $y$, and $z$ are non-zero integers. Then at least one of $x,y,z$ is even. And $2$ doesn't work. So at least one of $x,y,z$ is non-prime.

Remark: It turns out that one of $x$ or $y$ must be even, say $y$. There is a nice representation for all such Pythagorean triples. They are all of the shape $x=a(s^2-t^2)$, $y=2ast$, $z=a(s^2+t^2)$, where $a,s,t$ are positive integers, and $s$ and $t$ are of opposite parity (one is even and the other is odd).

Answer 2

They can't all be prime numbers. Let $A,B,C$ be sides of a right triangle and assume sides $A$ and $B$ were primes greater than 2, Then $A^2$ and $B^2$ are odd since $A$ and $B$ are greater than 2. So $C^2$ must be even thus $C$ is even.

Also, we can generate pythagorean numbers with integers $p$ and $q$ such that the sides are $A=2pq,B=p^2−q^2, C=p^2+q^2$ such that p and q are of different parity and $\gcd(p,q)=1$. Clearly Side $A$ is even and can't be 2 since if it were, then $p=1, q=1$ then side $B$ would be $0$, so they can not all be primes.

Answer 3

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 $11$, $60$, $61$.