Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$r\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer. A positive integer n is called r-full if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p^r$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> divides n whenever p is a prime divisor of n. Let q <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(|q|\ge 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>q</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an integer. In this paper, we prove irrationality of the infinite series <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sum _{n:r-{\text {full}}}a(n)/q^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>:</mo> <mi>r</mi> <mo>-</mo> <mtext>full</mtext> </mrow> </msub> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\{a(n)\}_{n\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is a sequence of integers satisfying suitable growth and non-zero conditions. As an application of main theorem, we derive linear independence over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> of the numbers <Equation ID="Equ62"> <MediaObject ID="MO1"> <ImageObject Color="BlackWhite" FileRef="MediaObjects/10998_2025_696_Equ62_HTML.png" Format="PNG" Height="112" Rendition="HTML" Resolution="300" Type="Linedraw" Width="720" /> </MediaObject> </Equation>

Irrationality and linear independence results for certain infinite series related to r-full numbers

期刊论文
发表日期：

2026年1月12日

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Jaroslav Hančl,
Florian Luca,
Yohei Tachiya

摘要

Let $r\ge 2$ $r \geq 2$ be an integer. A positive integer n is called r-full if $p^r$ $p^{r}$ divides n whenever p is a prime divisor of n. Let q $(|q|\ge 2)$ $(| q | \geq 2)$ be an integer. In this paper, we prove irrationality of the infinite series $\sum _{n:r-{\text {full}}}a(n)/q^n$ $\sum_{n : r - full} a (n) / q^{n}$ , where $\{a(n)\}_{n\ge 1}$ ${a (n)}_{n \geq 1}$ is a sequence of integers satisfying suitable growth and non-zero conditions. As an application of main theorem, we derive linear independence over $\mathbb {Q}$ $Q$ of the numbers