We investigate arithmetic properties of the sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\varvec{b(n)}}=\pmb {\mathcal {B}}_{M, N}\pmod {(n)}\,({\textbf {mod}} \;{\varvec{M}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">b</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">n</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mrow> <mi mathvariant="bold-script">B</mi> </mrow> <mrow> <mi>M</mi> <mo>,</mo> <mi>N</mi> </mrow> </msub> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">mod</mi> <mspace width="0.277778em" /> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> obtained from the base <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\varvec{M}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation> to base <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\varvec{N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">N</mi> </mrow> </math></EquationSource> </InlineEquation> shift map <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\pmb {\mathcal {B}}_{{\varvec{M,N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-script">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">M</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We prove that b(n) is purely periodic exactly when every prime divisor of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\varvec{M}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">M</mi> </mrow> </math></EquationSource> </InlineEquation> also divides <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\varvec{N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">N</mi> </mrow> </math></EquationSource> </InlineEquation>, and we explicitly determine the exact minimal period for all such cases. When <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\textbf {gcd}}\;{\varvec{(M,N)}}={\varvec{1}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">gcd</mi> <mspace width="0.277778em" /> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">M</mi> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">N</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mn mathvariant="bold">1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\varvec{b(n)}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">b</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">n</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\varvec{(N=1)}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold-italic">N</mi> <mo mathvariant="bold">=</mo> <mn mathvariant="bold">1</mn> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

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

期刊论文
发表日期：

2026年7月21日

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Ma Wanli

摘要

We investigate arithmetic properties of the sequence ${\varvec{b(n)}}=\pmb {\mathcal {B}}_{M, N}\pmod {(n)}\,({\textbf {mod}} \;{\varvec{M}})$ $b (n) = B_{M, N} (mod (n)) (\mod M)$ obtained from the base ${\varvec{M}}$ $M$ to base ${\varvec{N}}$ $N$ shift map $\pmb {\mathcal {B}}_{{\varvec{M,N}}}$ $B_{M, N}$ . We prove that b(n) is purely periodic exactly when every prime divisor of ${\varvec{M}}$ $M$ also divides ${\varvec{N}}$ $N$ , and we explicitly determine the exact minimal period for all such cases. When ${\textbf {gcd}}\;{\varvec{(M,N)}}={\varvec{1}}$ $\gcd (M, N) = 1$ , ${\varvec{b(n)}}$ $b (n)$ 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 ${\varvec{(N=1)}}$ $(N = 1)$ .