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October 13, 2025Open Access

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

Key Points

The sequence b(n) exhibits periodicity linked to prime divisors, providing a new viewpoint on digit maps.
When prime divisors of M don't divide N, b(n) yields additional solutions to the Prouhet-Tarry-Escott problem.
Finite-difference identities were utilized to evaluate multivariate polynomial sums, extending classical identities.
The minimal period of b(n) is determined precisely for cases involving prime powers, enhancing understanding of its behavior.

Abstract

We investigate arithmetic properties of the sequence b (n) = BMN (n) mod M obtained from the base-M to base-N shift map BMN. 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).

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Cite This Study

Weihong Ma (Sun,) studied this question.

synapsesocial.com/papers/68ecfebf950606aabec09410 https://doi.org/https://doi.org/10.48550/arxiv.2509.11269