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arXiv · 2510.01572

Elementary Proofs and Generalizations of Recent Congruences of Thejitha and Fathima

James A. Sellers
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Abstract

Motivated by recent work of Hirschhorn and the author, Thejitha and Fathima recently considered arithmetic properties satisfied by the function $a_5(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of five colors. They proved two sets of Ramanujan--like congruences satisfied by $a_5(n)$, relying heavily on modular forms. In this note, we prove their results via purely elementary means, utilizing generating function manipulations and elementary $q$-series dissections. We then extensively generalize these two sets of congruences to infinite families of divisibility properties in which the results of Thejitha and Fathima are specific instances.

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James A. Sellers. 2025-10-02. Elementary Proofs and Generalizations of Recent Congruences of Thejitha and Fathima. https://arxiv.org/abs/2510.01572

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