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All residues modulo (n+1)^2

Source: India IMOTC 2024 Day 2 Problem 1

May 31, 2024
abstract algebranumber theory

Problem Statement

Let nn be a positive integer. Let s:N→{1,…,n}s: \mathbb N \to \{1, \ldots, n\} be a function such that nn divides m−s(m)m-s(m) for all positive integers mm. Let a0,a1,a2,…a_0, a_1, a_2, \ldots be a sequence such that a0=0a_0=0 and ak=ak−1+s(k) for all k≥1.a_{k}=a_{k-1}+s(k) \text{ for all }k\ge 1. Find all nn for which this sequence contains all the residues modulo (n+1)2(n+1)^2.
Proposed by N.V. Tejaswi
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