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p-length vectors

Source: Kürchák 2018 P2

October 8, 2018
vectoranalytic geometrynumber theoryprime numbers

Problem Statement

Given a prime number pp and let v1‾,v2‾,…,vn‾\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n} be nn distinct vectors of length pp with integer coordinates in an R3\mathbb{R}^3 Cartesian coordinate system. Suppose that for any 1⩽j<k⩽n1\leqslant j<k\leqslant n, there exists an integer 0<ℓ<p0<\ell <p such that all three coordinates of vj‾−ℓ⋅vk‾\overline{v_j} -\ell \cdot \overline{v_k} is divisible by pp. Prove that n⩽6n\leqslant 6.
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