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<p>In this paper, we consider the two squares problem in order to introduce a combinatorial interpretation for Jacobi’s cubic analog of Euler’s pentagonal number theorem. Under certain conditions imposed by Fermat’s theorem on representations of integers as a sum of two squares, we derive a linear homogeneous recurrence relation for Euler’s partition function. In this context, we introduce two infinite conjectural families of Ramanujan type congruences and prove several special cases.</p>

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Jacobi’s cubic analog of the pentagonal number theorem and representations of \(24n+5\) as a sum of two squares

  • 期刊论文

  • 发表日期：

    2025年1月29日

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  • Cristina Ballantine,
  • Mircea Merca

摘要

In this paper, we consider the two squares problem in order to introduce a combinatorial interpretation for Jacobi’s cubic analog of Euler’s pentagonal number theorem. Under certain conditions imposed by Fermat’s theorem on representations of integers as a sum of two squares, we derive a linear homogeneous recurrence relation for Euler’s partition function. In this context, we introduce two infinite conjectural families of Ramanujan type congruences and prove several special cases.