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

2025/10/24 by Davis, Donald M.
#11A15 #11B50 #11E95 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.22078

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.

Citations

• Binomial coefficients involving infinite powers of primes

Related

• A study of a curious arithmetic function
• Binomial coefficients involving infinite powers of primes
• On the 2-adic valuation of (a^b-c^d)
• Numerical evidence toward a 2-adic equivariant "main conjecture"
• 2-adic Stirling functions and their zeros