Some 2-adic integers related to the odd part of 2^e!

TL;DR

The paper addresses the 2-adic limit $z=\lim_{e\to\infty} \operatorname{od}(2^e!)$ and its relation to the stable/unstable bits observed in $\operatorname{od}(2^e!)$, formalizing two conjectures from OEIS A359349. It introduces two 2-adic integers $z$ and $w$, a related constant $K$, and proves that there exists $K$ with $\operatorname{uns}(e,d)+K\equiv \operatorname{stab}(e+1,d)$ for all $d,e>d$, ultimately showing $K=-zw$. The key contributions include an explicit construction of $z$, the identity $K=-zw$, and strengthened congruence results that yield an efficient computation of $K$ via a refined analysis of $h(m)=\operatorname{odpr}(2^{m-1}+1,2^m-1)$ modulo $2^B$. The work also proves a more general product-congruence framework (Theorem hard) and uses it to establish Theorem thm2 through two technical lemmas about symmetric sums, thereby generalizing the original conjectures. This advances the understanding of 2-adic behavior in factorials and provides practical tools for computing bits of $z$.

Abstract

The odd part of 2^e! as e approaches infinity leads to a 2-adic integer z. The bits of z were publicized in OEIS-A359349, where two conjectures were made, relevant to computing z. We prove both of those conjectures. A second 2-adic integer, the limit of ((2^e-1)!!-1)/2^e, plays a key role in one proof.

Theorems & Definitions (39)

• Theorem 1.1
• Example 1.2
• Proposition 1.3
• proof
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3