Two binomial coefficient conjectures

TL;DR

This paper introduces two conjectures related to binomial coefficients involving prime powers and composites, focusing on their behavior modulo 16 and the value of Binomial[n, 2p] modulo n, expanding understanding beyond prime cases.

Contribution

It proposes two new conjectures addressing binomial coefficients with prime powers and composite numbers, an area less explored compared to prime-related coefficients.

Findings

Proposes a conjecture on counting binomial coefficients modulo 16.

Proposes a conjecture on the value of Binomial[n, 2p] modulo n.

Abstract

Much is known about binomial coefficients where primes are concerned, but considerably less is known regarding prime powers and composites. This paper provides two conjectures in these directions, one about counting binomial coefficients modulo 16 and one about the value of Binomial[n, 2p] modulo n.