Ulam numbers have zero density

TL;DR

This paper proves that the set of Ulam numbers has zero natural density, meaning they become negligible compared to all natural numbers as numbers grow larger.

Contribution

It establishes that the natural density of Ulam numbers is zero, providing a rigorous mathematical proof of their sparse distribution.

Findings

Ulam numbers have zero natural density.

The proportion of Ulam numbers among natural numbers tends to zero.

Ulam numbers are asymptotically negligible in the set of natural numbers.

Abstract

In this paper, we show that the natural density $\mathcal{D}[(U_m)]$ of Ulam numbers $(U_m)$ satisfies $\mathcal{D}[(U_m)]=0$. That is, we show that for $(U_m)\subset [1,k]$ \begin{align} \lim \limits_{k\longrightarrow \infty}\frac{\left |(U_m)\cap [1,k]\right |}{k}=0.\nonumber \end{align}