A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities

2025/09/14 by Ma, Wanli
#11A63 #11B83 #11P05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.11269

Abstract

We investigate arithmetic properties of the sequence b(n) = B_MN(n) mod M obtained from the base-M to base-N shift map B_MN.We prove that b(n) is ultimately periodic exactly when every prime divisor of M also divides N; in that case we bound (and, for prime powers, determine) the minimal period.When the condition fails, b(n) supplies new solutions to the Prouhet-Tarry-Escott problem.To analyze this situation we introduce a family of finite-difference identities and use them to evaluate two weighted multivariate polynomial sums, thereby extending identities that arise from the classical sum-of-digits function (N=1).

Citations

• BASE SHIFTING SEQUENCES AND THE PALINDROME PROPERTY
• Rarefied Thue-Morse Sums Via Automata Theory and Logic
• Rudin-Shapiro Sums Via Automata Theory and Logic
• Settling some sum suppositions
• A New Proof of the Prouhet-Tarry-Escott Problem
• Some identities involving Prouhet-Thue-Morse sequence and its relatives

Related

• On the Iterates of Digit Maps
• Periodic Sequences modulo m
• Arithmetic Properties of Periodic Maps
• Congruence properties of binary partition functions
• Periodic minimum in the count of binomial coefficients not divisible by a prime