A Simple Proof of Schmidt's Conjecture

TL;DR

This paper provides a new, simplified proof confirming Asmus Schmidt's conjecture that certain sequence coefficients, defined through binomial sums, are always integers for all positive integers r.

Contribution

The paper introduces a straightforward proof establishing the integrality of the sequence coefficients c^{(r)}_k, confirming Schmidt's conjecture.

Findings

Confirmed all c^{(r)}_k are integers for r ≥ 1

Provided a simplified proof of Schmidt's conjecture

Enhanced understanding of binomial coefficient sequences

Abstract

For any integer $r \geq 1$, the sequence of numbers $\{{c^{(r)}_{k}}\}_{k \geq 0} $ is defined implicitly by [\sum_k\binom{n}{k}^r\binom{n+k}{k}^r = \sum_k\binom{n}{k}\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,...] Asmus Schmidt conjectured that all $c^{(r)}_k$ are integers. We give a new proof of this fact.