MathDB
Easy NT with a ring

Source: Romanian District Olympiad 2024 12.3

March 10, 2024
number theoryRing Theory

Problem Statement

Let kk be a positive integer. A ring (A,+,⋅)(A,+,\cdot) has property PkP_k if for any a,b∈Aa,b\in A there exists c∈Ac\in A such that ak=bk+ck.a^k=b^k+c^k. [*]Give an example of a finite ring (A,+,⋅)(A,+,\cdot) which does not have PkP_k for any k⩾2.k\geqslant 2. [*]Let n⩾3n\geqslant 3 be an integer and Mn={m∈N:(Zn,+,⋅) has Pm}.M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}. Prove that all the elements of MnM_n are odd integers and that (Mn,⋅)(M_n,\cdot) is a monoid.
Back to ProblemsView on AoPS