[Paper Review] Congruences for the Number of Cubic Partitions Derived from Modular Forms

William Y. C. Chen, Bernard L. S. Lin|ArXiv.org|Oct 7, 2009

Advanced Mathematical Identities22 references18 citations

TL;DR

This paper establishes new congruence relations for the number of cubic partitions $ a(n) $ using modular forms, proving $ a(25n+22) \equiv 0 \pmod{5} $ and $ a(49n+15) \equiv a(49n+29) \equiv a(49n+36) \equiv a(49n+43) \equiv 0 \pmod{7} $. It further shows that $ a(n) $ takes infinitely many even and infinitely many odd values, extending Ramanujan-type arithmetic properties to cubic partition functions via modular form theory.

ABSTRACT

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has several analogous properties to the number p(n) of partitions, including the generating function, the continued fraction, and congruence relations. To be more specific, we show that $a(25n+22)\equiv 0 ({ m mod} 5)$, $a(49n+15) \equiv a(49n+29) \equiv a(49n+36) \equiv a(49n+43) \equiv 0 ({ m mod} 7)$. Furthermore, we prove that a(n) takes infinitely many even values and infinitely odd values.

Motivation & Objective

• To extend Ramanujan's classical partition congruences to the context of cubic partitions.
• To derive new arithmetic properties of $ a(n) $, the number of cubic partitions, using the theory of modular forms.
• To establish that $ a(n) $ takes infinitely many even and infinitely many odd values, analogous to the partition function $ p(n) $.
• To prove specific congruences modulo 5 and 7 for $ a(n) $, inspired by Ramanujan's work on $ p(n) $.

Proposed method

• Constructing modular forms of weight 6 for the congruence subgroup $ \Gamma_0(4802) $ to analyze $ a(n) $.
• Using the generating function $ \sum_{n=0}^\infty a(n)q^n = \frac{1}{(q;q)_\infty (q^2;q^2)_\infty} $ as the foundation for modular analysis.
• Applying the method of Hirschhorn and Hunt to derive congruences modulo 5 and 7 by manipulating modular forms and their Fourier coefficients.
• Verifying congruences modulo 7 by checking finitely many initial coefficients, leveraging the modularity of the involved forms.
• Deriving a recurrence relation modulo 2: $ a(n) + \sum_{0 < k+k^2 \leq n} a(n-k-k^2) \equiv \Delta(n) \pmod{2} $, where $ \Delta(n) = 1 $ if $ n $ is a triangular number.
• Using contradiction and properties of triangular numbers to prove that $ a(n) $ assumes both even and odd values infinitely often.

Experimental results

Research questions

• RQ1 Do cubic partition numbers $ a(n) $ satisfy congruences analogous to Ramanujan's partition congruences modulo 5 and 7?
• RQ2 Can modular forms be used to prove new arithmetic properties of $ a(n) $, such as congruences modulo prime powers?
• RQ3 Does $ a(n) $ exhibit infinite parity variation, i.e., take both even and odd values infinitely often?
• RQ4 Is there a combinatorial or modular explanation for the congruence $ a(3n+2) \equiv 0 \pmod{3} $, and can this be extended to higher moduli?

Key findings

• The paper proves $ a(25n+22) \equiv 0 \pmod{5} $, establishing a new Ramanujan-type congruence for cubic partitions modulo 5.
• It establishes four congruences modulo 7: $ a(49n+15) \equiv a(49n+29) \equiv a(49n+36) \equiv a(49n+43) \equiv 0 \pmod{7} $.
• The function $ a(n) $ takes infinitely many even values and infinitely many odd values, as shown via a recurrence modulo 2 and contradiction arguments.
• The proof of the modulo 7 congruences relies on verifying finitely many initial coefficients (up to $ n \leq 587 $) due to modularity and level structure of the involved modular forms.
• The generating function for $ a(n) $, $ \sum a(n)q^n = \frac{1}{(q;q)_\infty (q^2;q^2)_\infty} $, is central to deriving both the congruences and the parity result.
• The recurrence $ a(n) + \sum_{0 < k+k^2 \leq n} a(n-k-k^2) \equiv \Delta(n) \pmod{2} $ is key to analyzing the parity of $ a(n) $, where $ \Delta(n) $ is 1 if $ n $ is triangular, 0 otherwise.