The only even prime number is 2, and that can't be the perimeter of a triangle
with consecutive integer sides, so the perimeter must be odd in order to be a
prime number.  So we cannot have the shortest side odd, the middle sized side
even,  and the largest side odd, since the perimeter would then be even,
since "odd+even+odd=even"

So the shortest side must be even, the medium-size side must be odd, and
the longest side must be even, and that works since even+odd+even=odd.

So let the shortest side be 2n where n is any integer,

So the sides are 2n, 2n+1, and 2n+2.  That makes the perimeter be:

2n + (2n+1) + (2n+2)

So perimeter = 2n + 2n + 1 + 2n + 2

Perimeter = 6n + 3

Perimeter = 3(2n+1)

So the perimeter is always divisible by 3 so it cannot be prime.

The answer is NO.

Edwin