On the Divisibility of 7-Elongated Plane Partition Diamonds by Powers of 8

TL;DR

This paper proves the existence of an infinite family of congruences for the 7-elongated plane partition function modulo powers of 8 using classical methods, confirming a conjecture and contrasting with more modern approaches.

Contribution

It establishes an infinite congruence family for d_7(n) modulo powers of 8 using classical polynomial techniques, unlike previous proofs requiring advanced methods.

Findings

Confirmed the conjecture of infinite congruences for d_7(n) modulo powers of 8.

Demonstrated classical proof techniques are sufficient for this problem.

Provided new insights into divisibility properties of plane partition functions.

Abstract

In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the $k$-elongated plane partition function $d_k(n)$ by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function $d_7(n)$. We prove that such a congruence family exists -- indeed, for powers of 8. The proof utilizes only classical methods, i.e., integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for $d_k(n)$ which require more modern methods to prove.