MathDB
Problem 1 - IMOR 2018

Source: 2nd IMOR - 2018

July 13, 2018
IMORnumber theory

Problem Statement

Let pp be a prime number, and XX be the set of cubes modulo pp, including 00. Denote by C2(k)C_2(k) the number of ordered pairs (x,y)∈X×X(x, y) \in X \times X such that x+y≡k(modp)x + y \equiv k \pmod p. Likewise, denote by C3(k)C_3(k) the number of ordered pairs (x,y,z)∈X×X×X(x, y, z) \in X \times X \times X such that x+y+z≡k(modp)x + y + z \equiv k \pmod p. Prove that there are integers a,ba, b such that for all kk not in XX, we have C3(k)=a⋅C2(k)+b. C_3(k) = a\cdot C_2(k) + b.
Proposed by Murilo Corato, Brazil.
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