Number theory · Perfect cubes · Coprime factors of a power · Unique factorization · Difference of cubes · Primality forces trivial factorization · Bounding between consecutive cubes
Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number
\[ k^3 + pk^2 \]
is a perfect cube. Prove that \(3 \mid p - 1\).
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Serbian National Competition (Drzavno takmicenje) 2013, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source