On Additive Representations of Integers by Binomial Coefficients

TL;DR

This paper explores how positive integers can be expressed as sums of binomial coefficients, providing explicit proofs for specific cases, comparing with polygonal numbers, and discussing general existence and bounds.

Contribution

It offers new elementary proofs for certain cases, clarifies why naive counting fails, and presents both conditional and unconditional results for general cases.

Findings

Elementary proofs for k=2 and k=3 cases

Comparison with polygonal number theory

Existence results and computational evidence

Abstract

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional cases, general existence results have received less explicit treatment. This paper provides: $\bullet$ explicit elementary proofs for the cases ($k=2$) and ($k=3$), $\bullet$ a comparison with classical polygonal number theory, $\bullet$ an explanation of why naive counting arguments fail for general ($k$), $\bullet$ conditional and unconditional existence results for general ($k$), $\bullet$ and a discussion of quantitative bounds and computational evidence. Together these give a unified and transparent framework for understanding additive representations by binomial coefficients.