MathDB

Perfect Cubes

Source: Indonesia Regional 2023 Essay Number 2

July 16, 2023

number theoryperfect cubesIndonesiaRMO

Problem Statement

Let

K

be a positive integer such that there exist a triple of positive integers

(x,y,z)

such that

x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx

are all perfect cubes. (a) Prove that

K \ne 2

and

K \ne 4

(b) Find the minimum value of

K

that satisfies.