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

TL;DR

This paper proves two conjectures related to the 2-adic integer derived from the odd part of factorial powers of 2, enhancing understanding of their 2-adic properties and sequences.

Contribution

It provides rigorous proofs for two conjectures about 2-adic integers associated with factorial powers of 2, clarifying their structure and sequence behavior.

Findings

Proof of two conjectures from OEIS-A359349

Identification of a second 2-adic integer limit involving double factorials

Enhanced understanding of 2-adic properties related to factorial sequences

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.