Prove that there exist infinitely many natural numbers \(a\) with the following property: the number \(z = n^4+a\) is not prime for any natural number \(n\).
Let \(a,b,c>0\) be the length of sides of a triangle. Show that the triangle is right-angled if and only if \((a^4+b^4+c^4)^2 = 2(a^8+b^8+c^8)\). Note that this is a symmetric characterization of right-angled triangles by its side lengths.