Abstract
Recently Amdeberhan, Sellers, and Singh introduced a new infinite family of partition functions called generalized cubic partitions. Given a positive integer d, they let \(a_d(n)\) be the counting function for partitions of n in which the odd parts are unrestricted and the even parts are d-colored. These partitions are natural generalizations of Chan’s notion of cubic partitions, as they coincide when \(d=2.\) Many Ramanujan-like congruences exist in the literature for cubic partitions, and in their work Amdeberhan, Sellers, and Singh proved a collection of congruences satisfied by \(a_d(n)\) for various \(d \ge 1\), including an infinite family with prime moduli. Our goal in this paper is to prove a family of congruences modulo powers of 5 for \(a_3(n)\). More specifically, our main theorem asserts
In order to prove these congruences, we use an approach centered around modular functions, as in the seminal work of Watson and Atkin on proving Ramanujan’s congruences for the partition function p(n). However, due to the complexity of the modular curve \(X_0(10)\) associated to our modular functions, the classical method cannot be directly applied. Rather, we utilize the very recently developed localization method of Banerjee and Smoot, which is designed to treat congruence families over more complicated modular curves, such as \(X_0(10).\)
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References
Berndt, B.C., Ono, K.: Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary. 42, 42–63 (1999). The Andrews Festschrift (Maratea, 1998)
Watson, G.N.: Ramanujans Vermutung über Zerfällungszahlen. J. Reine Angew. Math. 179, 97–128 (1938). https://doi.org/10.1515/crll.1938.179.97
Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glasgow Math. J. 8, 14–32 (1967). https://doi.org/10.1017/S0017089500000045
Smoot, N.A.: A congruence family for 2-elongated plane partitions: an application of the localization method. J. Number Theory 242, 112–153 (2023). https://doi.org/10.1016/j.jnt.2022.07.014
Banerjee, K., Smoot, N.A.: The localization method applied to \(k\)-elongated plane partitions and divisibility by 5. Math. Z. 309(3), 46–51 (2025). https://doi.org/10.1007/s00209-024-03672-9
Banerjee, K., Smoot, N.: 2-Elongated plane partitions and powers of 7: the localization method applied to a genus 1 congruence family. Can. J. Math. (2025). https://doi.org/10.4153/S0008414X25101120
Amdeberhan, T., Sellers, J.A., Singh, A.: Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime. Aequ. Math. 99(3), 1197–1208 (2025). https://doi.org/10.1007/s00010-024-01116-7
Chan, H.-C.: Ramanujan’s cubic continued fraction and an analog of his “most beautiful identity’’. Int. J. Number Theory 6(3), 673–680 (2010). https://doi.org/10.1142/S1793042110003150
Chan, H.H., Toh, P.C.: New analogues of Ramanujan’s partition identities. J. Number Theory 130(9), 1898–1913 (2010). https://doi.org/10.1016/j.jnt.2010.02.017
Chern, S., Dastidar, M.G.: Congruences and recursions for the cubic partition. Ramanujan J. 44(3), 559–566 (2017). https://doi.org/10.1007/s11139-016-9852-7
Guadalupe, R.: A note on congruences for generalized cubic partitions modulo primes. Integers 25, 20–5 (2025). https://doi.org/10.5281/zenodo.14907350
Diamond, F., Shurman, J.: A First Course in Modular Forms. Graduate Texts in Mathematics, vol. 228, p. 436. Springer, New York (2005)
Atkin, A.O.L., Lehner, J.: Hecke operators on \(\Gamma _{0}(m)\). Math. Ann. 185, 134–160 (1970). https://doi.org/10.1007/BF01359701
Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and \(q\)-series. In: CBMS Regional Conference Series in Mathematics, vol. 102, p. 216. American Mathematical Society, Providence, RI (2004)
Radu, C.-S.: An algorithmic approach to Ramanujan-Kolberg identities. J. Symb. Comput. 68, 225–253 (2015). https://doi.org/10.1016/j.jsc.2014.09.018
Sturm, J.: On the congruence of modular forms. In: Number Theory (New York, 1984–1985). Lecture Notes in Mathematics, vol. 1240, pp. 275–280. Springer, Berlin (1987). https://doi.org/10.1007/BFb0072985
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The author would like to sincerely thank Marie Jameson for her invaluable advice and guidance. The author is also grateful to the anonymous referees for their helpful suggestions that enhanced this manuscript.
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Initial relations for the proof of Theorem 4.5
Initial relations for the proof of Theorem 4.5
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Dockery, D. A congruence family modulo powers of 5 for generalized cubic partitions via the localization method. Ramanujan J 69, 65 (2026). https://doi.org/10.1007/s11139-026-01344-2
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DOI: https://doi.org/10.1007/s11139-026-01344-2