Prime Square

Relevant wiki: Perimeter

Let $x$ and $y$ be the side lengths of the rectangle. Hence, the square's side length = $\dfrac{x+y}{2}$ .

Then the rectangle's area = $xy$ , and the square's area = $\dfrac{(x+y)^2}{4}$ .

Thus, $\dfrac{(x+y)^2}{4} - xy = \dfrac{(x^2 +2xy +y^2)-4xy }{4} = \dfrac{(x-y)^2}{4} = 4$ .

Taking square roots both sides, we will get: $\dfrac{|x-y|}{2} = 2$ . $|x - y| = 4$ .

In other words, supposedly if $x$ is the shorter side length of the rectangle, then $x+2$ is the side length of the square and $x+4$ is the longer side length of the rectangle.

Since all these lengths are prime integers, only the triple $3 , 5 , 7$ applies because in the higher triple range, one of the three numbers will fall upon the multiple of $3$ .

As a result, the square's perimeter = $4\times 5 = \boxed{20}$ .

Alternatively, the rectangle's perimeter = $2\times(3+7) = \boxed{20}$ .