<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1997_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <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>. Nayaka, Dharmendra and Kumar proved some congruences modulo 8, 16 and 32 for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1997_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <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>. Recently, Saikia and Sarma established some congruences modulo 64 for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1997_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <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> by using both elementary techniques and the theory of modular forms. In their paper, they also posed two conjectures on infinite families of congruences modulo 64 and 128 for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1997_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <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>. In this paper, we confirm the two conjectures.</p>

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Proofs of two conjectures on infinite families of congruences of overcubic partition triples

  • Jiayu Chen,
  • Jing Jin,
  • Olivia X. M. Yao

摘要

Let \(\overline{bt}(n)\) bt ¯ ( n ) denote the number of overcubic partition triples of n. Nayaka, Dharmendra and Kumar proved some congruences modulo 8, 16 and 32 for \(\overline{bt}(n)\) bt ¯ ( n ) . Recently, Saikia and Sarma established some congruences modulo 64 for \(\overline{bt}(n)\) bt ¯ ( n ) by using both elementary techniques and the theory of modular forms. In their paper, they also posed two conjectures on infinite families of congruences modulo 64 and 128 for \(\overline{bt}(n)\) bt ¯ ( n ) . In this paper, we confirm the two conjectures.