Two-Prime Triples

A Pythagorean triple is a triple of positive integers \((x,y,z)\) such that \(x^2+y^2=z^2.\) Find the sum of all \(z\leq 100\), for which there exists a Pythagorean triple \((x,y,z)\) with exactly two of the numbers \(x\), \(y\) and \(z\) being prime.

Details and assumptions

Each distinct value of \(z\) should only be included once. For example, \( (3, 4, 5)\) and \((4, 3, 5)\) satisfy the conditions, and so \(z=5\) is a possible value.

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