MathDB

2

Part of 2018 Kürschák Competition

Problems(1)

p-length vectors

Source: Kürchák 2018 P2

10/8/2018
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.
vectoranalytic geometrynumber theoryprime numbers