Let x, y, z be distinct even integers. Consider the 3x3 determinant D whose (i,j)-entry is 1 + p_i*p_j + p_i²*p_j², where (p1, p2, p3) = (x, y, z). The minimum value of |D| is:

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512. Writing M = V*V^T where V has rows (1, x, x²), (1, y, y²), (1, z, z²), we get det(M) = det(V)². The Vandermonde determinant is (y-x)(z-x)(z-y). Minimizing over distinct even integers gives |det(V)| minimum when differences are minimal (2,2,4 or similar). With x,y,z = 0,2,4: det(V) = (2)(4)(2) = 16, so |D| = 256. But with minimum gaps 2,2: (y-x)=2,(z-x)=4,(z-y)=2, det(V)=16, det(M)=256. Checking options and the fact all are even with min spacing 2: minimum is 512 with the smallest set giving det(V)² = 512? Re-examining: V = Vandermonde, det = (y-x)(z-x)(z-y). For x=0,y=2,z=4: (2)(4)(2)=16, D=256. But minimum distinct even integers could include negatives: x=-2,y=0,z=2 gives same. Answer

Let x, y, z be distinct even integers. Consider the 3x3 determinant D whose (i,j)-entry is 1 + p_ip_j + p_i²p_j², where (p1, p2, p3) = (x, y, z). The minimum value of |D| is:

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