🔢 mathematics

Some new congruences on biregular overpartitions

This paper establishes infinitely many families of new congruences modulo 3 and powers of 2 for biregular overpartitions with specific parameter pairs, including general cases for $(5,2^t)$ , $(3,2^t)$ , and $(4,3^t)$ , by utilizing the theory of Hecke eigenforms, a Newman identity, and dissection formulas.

Original authors: N. K. Meher

Published 2026-07-31

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: N. K. Meher

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where numbers aren't just tools for counting your allowance or tracking your video game score, but are instead the bricks and mortar of a vast, invisible city. This city is the realm of number theory, a branch of mathematics that studies the hidden patterns and relationships between whole numbers. In this city, there's a fascinating game called "partitioning." Imagine you have a pile of 4 identical Lego bricks. You can stack them in a single tower, split them into two towers of 2, or make a tower of 3 and a single brick. Each way you arrange them is a "partition." Mathematicians have spent centuries counting these arrangements, but they've recently started playing a more complex version of the game called "overpartitions." In this version, the very first time a number appears in your stack, you can put a little hat on it (or "overline" it). This tiny hat doubles the number of possible arrangements, making the city much more crowded and the patterns much harder to find.

Now, imagine you're a detective trying to find specific clues in this crowded city. You're looking for arrangements that follow strict rules: maybe no tower can be a multiple of 3, or maybe no tower can be a multiple of 5. These are called "regular" partitions. When you combine rules—like "no multiples of 3 AND no multiples of 5"—you get "biregular" overpartitions. The big question in this corner of math is: Do these numbers follow a secret rhythm? Specifically, if you count these special arrangements, do they always leave a remainder of 0 when divided by 3, or 8, or some other number? Finding these "congruences" is like discovering that every 10th house in the city has a blue door, or that every 7th street is always empty. It reveals a deep, underlying order in what looks like chaos.

This paper is a treasure map drawn by mathematician N.K. Meher, leading us to new, hidden patterns in the world of biregular overpartitions. While previous detectives had already found some blue doors in specific neighborhoods (like when the forbidden numbers were 4 and 3), Meher has expanded the search to new territories. The paper focuses on pairs of forbidden numbers, such as (2, 9), (5, 2), and (5, 4), and even general families like (5, 2t) where t is any number 3 or bigger.

Using a powerful toolkit that includes "generating functions" (which are like magical recipes that turn a list of numbers into a single, flowing equation), "dissection formulas" (a way of slicing those equations into smaller, manageable pieces), and the theory of "Hecke eigenforms" (a high-level mathematical structure that acts like a fingerprint for these patterns), Meher proves that these new neighborhoods are full of secrets. The paper doesn't just guess; it proves with absolute certainty that for these specific pairs of rules, there are infinitely many families of numbers that are perfectly divisible by 3 or powers of 2.

For instance, the paper shows that if you look at the number of ways to arrange your overpartitions under the rule "no multiples of 2 or 9," you will find that the count for certain numbers (like 6n + 3) is always divisible by 4, and for others (like 6n + 5) is always divisible by 8. It's as if the city has a strict zoning law that forces certain blocks to be empty or perfectly divisible. The author also discovers that these patterns repeat in a multiplicative way: if you take a number that fits the pattern and multiply it by a specific prime number (like 5), the new number also fits the pattern, often with a predictable change.

The paper is a rigorous proof, meaning these aren't just lucky guesses or computer simulations; they are mathematical facts derived from logic. Meher establishes that for pairs like (5, 2t) where t is 3 or more, and (3, 2t) or (4, 3t) for any t, there are infinite sequences of numbers where the count of these special arrangements vanishes modulo 3 or 8. It's a significant expansion of our knowledge, showing that the hidden rhythm of these overpartitions is even more complex and widespread than we previously knew, revealing new "blue doors" in the infinite city of numbers.

Technical Summary: Some New Congruences on Biregular Overpartitions

Problem Statement
The paper investigates the arithmetic properties of $\bar{B}_{\ell_1, \ell_2}(n)$ , the number of $(\ell_1, \ell_2)$ -biregular overpartitions of $n$ . An overpartition is a partition where the first occurrence of each part may be overlined. A biregular overpartition is one where no part is divisible by either $\ell_1$ or $\ell_2$ , with the constraints $\gcd(\ell_1, \ell_2)=1$ and $\ell_1, \ell_2 > 1$ . While recent works by Nadji, Ahmia, and Ramírez [18]; Alanazi, Munagi, and Saikia [1]; and Paudel, Sellers, and Wang [21] have established congruences for specific pairs such as $(4,3)$ , $(4,9)$ , and $(8,3)$ , this paper aims to extend these results to new pairs and general families of parameters.

Methodology
The author employs a combination of classical $q$ -series manipulations, dissection formulas, and the theory of modular forms. The primary tools include:

Generating Functions: The generating function for $\bar{B}_{\ell_1, \ell_2}(n)$ is expressed as an eta-quotient (Equation 1.1).
Dissection Formulas: The paper utilizes various dissection identities (Lemmas 2.5–2.10) derived from Ramanujan's work and subsequent literature to separate terms in the generating function based on the index modulo specific integers (e.g., extracting terms $q^{2n}$ , $q^{3n}$ , etc.).
Modular Forms and Hecke Eigenforms: The author identifies specific eta-quotients as modular forms in spaces $M_k(\Gamma_0(N), \chi)$ . By proving these forms are Hecke eigenforms, the paper leverages the properties of Hecke operators $T_p$ to derive recurrence relations for the coefficients.
Newman's Identity: For specific cases, the paper applies an identity due to Newman [19] regarding the coefficients of products of eta functions to establish congruences for primes $p \equiv 1 \pmod m$ .
Congruence Properties: Lemma 2.8, a standard congruence for powers of $f_m$ modulo $p^k$ , is used repeatedly to simplify expressions modulo 3 and powers of 2.

Key Contributions and Results
The paper establishes infinitely many families of congruences modulo 3 and powers of 2 for the following pairs $(\ell_1, \ell_2)$ :

Specific Pairs: $(2, 9)$ , $(5, 2)$ , $(5, 4)$ , $(8, 3)$ , and $(5, 8)$ .
General Families:
- $(5, 2^t)$ for all $t \geq 3$ .
- $(3, 2^t)$ for all $t \geq 1$ .
- $(4, 3^t)$ for all $t \geq 1$ .

Specific Findings by Section:

Section 3 ( $\bar{B}_{2,9}(n)$ ): The author proves that $\bar{B}_{2,9}(6n+3) \equiv 0 \pmod 4$ and $\bar{B}_{2,9}(6n+5) \equiv 0 \pmod 8$ . Using Hecke eigenform theory, infinite families of congruences modulo 8 are derived for arguments of the form $6p_1^2 \dots p_{k+1}^2 n + \dots$ where primes $p_i \geq 5$ and $p_i \not\equiv 1 \pmod 6$ . Multiplicative formulas modulo 8 and 3 are also established.
Section 4 ( $\bar{B}_{5,2^t}(n)$ ): For $t \geq 3$ , the paper shows $\bar{B}_{5,2^t}(4n+3) \equiv 0 \pmod 8$ . It derives infinite families of congruences modulo 8 for arguments involving squares of primes $p \not\equiv 1 \pmod 4$ . Multiplicative formulas involving a function $f(p)$ dependent on $p \pmod{20}$ are presented.
Sections 5 & 6 ( $\bar{B}_{5,2}(n)$ and $\bar{B}_{5,4}(n)$ ): Similar congruences modulo 4 are established for these specific cases, including $\bar{B}_{5,2}(4n+3) \equiv 0 \pmod 4$ and $\bar{B}_{5,4}(4n+3) \equiv 0 \pmod 4$ .
Section 7 ( $\bar{B}_{8,3}(n)$ ): The paper proves $\bar{B}_{8,3}(36n+33) \equiv 0 \pmod 3$ .
Section 8 ( $\bar{B}_{4,3^t}(n)$ ): For all $t \geq 1$ , the author establishes $\bar{B}_{4,3^t}(3n) \equiv 0 \pmod 8$ (for $n \geq 1$ ) and various congruences modulo 4 for arguments $6n+4$ , $12n+7$ , and $3n+2$ .
Section 9 ( $\bar{B}_{3,2^t}(n)$ ): For all $t \geq 1$ , the paper proves $\bar{B}_{3,2^t}(16n+6) \equiv \bar{B}_{3,2^t}(16n+10) \equiv \bar{B}_{3,2^t}(16n+14) \equiv 0 \pmod 8$ .

Significance
The paper claims to extend the existing literature on biregular overpartitions by providing new infinite families of congruences for pairs $(\ell_1, \ell_2)$ that were not previously covered or fully generalized. By utilizing the theory of Hecke eigenforms and Newman's identity, the author moves beyond specific computational examples to establish structural properties of these partition functions. The results contribute to the broader understanding of the arithmetic behavior of overpartitions with restricted part divisibility, specifically highlighting patterns modulo 3 and powers of 2. The work relies on the foundational theories of modular forms and $q$ -series, offering a rigorous algebraic framework for these new congruences.

More like this