A curve is defined parametrically by the equations
\[ x = t^2 - 4t + 7 \]
\[ y = 3t + 5 \]
Find the value of \( t \) for which the Cartesian form of the curve is a parabola that intersects the x-axis at a prime number. Identify the prime number and show that the difference of squares of the coordinates of the intersection point is a positive integer. Let's think step by step and output the final answer within \boxed{}.
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