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Determinant divisible by p^3

Source: VJIMC 2018, Category I, Problem 2

April 14, 2018
number theoryprime numbers

Problem Statement

Find all prime numbers pp such that p3p^3 divides the determinant ∣2211…11321…111421⋮⋮⋱111(p+7)2∣.\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.
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