Article

Certain Diophantine equations and new parity results for 21-regular partitions

January 2023
Acta Arithmetica 210

DOI:10.4064/aa230203-5-7

Authors:

University of Virginia

Indian Institute of Technology Guwahati

Indian Institute of Technology Guwahati

Arithmetic properties of 5-regular partitions into distinct parts

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Full-text available

Nov 2024

A partition is said to be

\ell

-regular if none of its parts is a multiple of

\ell

. Let

b^\prime_5(n)

denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of

b^\prime_5(n)

. We provide full characterization of the parity of

b^\prime_5(2n+1)

, present several congruences modulo 4, and prove that the generating function of the sequence

(b^\prime_5(5n+1))

is lacunary modulo any arbitrary positive powers of 5.

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Parity of 3-regular partition numbers and Diophantine equations

December 2022

Let

b_3(n)

be the number of 3-regular partitions of n. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo 2 for

b_3(2n)

involving every prime p with

p \equiv 13, 17, 19, 23 \pmod {24}

, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo 2 for

b_3(2n)

involving every prime

p\geqslant 5

. In this ... [Show full abstract] paper, we introduce new infinite Ramanujan type congruences modulo 2 for

b_3(2n)

. They complement naturally the results of Keith-Zanello and Yao and involve primes in

\mathcal P=\{p \text{ prime } : \exists \, j\in \{1,4,8\},\, x, y \in \mathbb Z,\, \gcd(x,y)=1 \text { with } x^2+216y^2=jp\}

whose Dirichlet density is 1/6. As a key ingredient in our proof we show that of the number of primitive solutions for

x^2+216y^2=pm

,

p \in \mathcal P

,

p\nmid m

and

pm\equiv 1\pmod{24}

, is divisible by 8. Here, the difficulty arises from the fact that 216 is not idoneal. We also give a conjectural exact formula for the number of solutions for this Diophantine equation. In the second part of the article, we study reversals of Euler-type identities. These are motivated by recent work of the second author on a reversal of Schur's identity which involves 3-regular partitions weighted by the parity of their length.

Preprint

Certain Diophantine equations and new parity results for 21-regular partitions

January 2023

For a positive integer

t\geq 2

, let

b_{t}(n)

denote the number of t-regular partitions of a nonnegative integer n. In a recent paper, Keith and Zanello investigated the parity of

b_{t}(n)

when

t\leq 28

. They discovered new infinite families of Ramanujan type congruences modulo 2 for

b_{21}(n)

involving every prime p with

p\equiv 13, 17, 19, 23 \pmod{24}

. In this paper, we ... [Show full abstract] investigate the parity of

b_{21}(n)

involving the primes p with

p\equiv 1, 5, 7, 11 \pmod{24}

. We prove new infinite families of Ramanujan type congruences modulo 2 for

b_{21}(n)

involving the odd primes p for which the Diophantine equation

8x^2+27y^2=jp

has primitive solutions for some

j\in\left\lbrace1,4,8\right\rbrace

, and we also prove that the Dirichlet density of such primes is equal to 1/6. Recently, Yao provided new infinite families of congruences modulo 2 for

b_{3}(n)

and those congruences involve every prime

p\geq 5

based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo 2 for

b_{21}(n)

, and these congruences imply that

b_{21}(n)

is odd infinitely often.

Last Updated: 22 Oct 2024