A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities
Number Theory
2025-09-16 v1
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).
Keywords
Cite
@article{arxiv.2509.11269,
title = {A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities},
author = {Wanli Ma},
journal= {arXiv preprint arXiv:2509.11269},
year = {2025}
}