<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{bt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of overcubic partition triples of <i>n</i>. By analyzing the generating functions of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{bt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> along specific arithmetic progressions and their successive differences, we establish several families of internal congruences modulo high powers of 2. One of these congruence families substantially generalizes recent results of Saikia and Sarma (2025) as well as Chen, Jin and Yao (2025). Moreover, we conjecture the existence of congruence relations modulo high powers of 2 for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{bt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, revealing a hidden 2-adic regularity in the internal structure of this partition function.</p>

Internal Congruences Modulo Powers of 2 for Overcubic Partition Triples

期刊论文
发表日期：

2026年3月31日

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Dazhao Tang

摘要

Let $\overline{bt}(n)$ $\bar{bt} (n)$ denote the number of overcubic partition triples of n. By analyzing the generating functions of $\overline{bt}(n)$ $\bar{bt} (n)$ along specific arithmetic progressions and their successive differences, we establish several families of internal congruences modulo high powers of 2. One of these congruence families substantially generalizes recent results of Saikia and Sarma (2025) as well as Chen, Jin and Yao (2025). Moreover, we conjecture the existence of congruence relations modulo high powers of 2 for $\overline{bt}(n)$ $\bar{bt} (n)$ , revealing a hidden 2-adic regularity in the internal structure of this partition function.