Solutions of the problem of Erd\"os-Sierpi\'nski: $\sigma(n)=\sigma(n+1)$

TL;DR

This paper reports the discovery of 1268 solutions to the Erdös-Sierpiński problem for n ≤ 1.5×10^{10}, explores empirical properties of these solutions, and presents related theorems and asymptotic behavior.

Contribution

The paper provides the first extensive computational solutions, empirical observations, and theoretical insights into the Erdös-Sierpiński problem.

Findings

All solutions' divisor sums are divisible by 6.

Solutions exhibit a pattern where multiple solutions share the same divisor sum.

The number of solutions up to n grows approximately as n^{1/3}.

Abstract

For $n\leq 1.5 \cdot 10^{10}$, we have found a total number of 1268 solutions to the Erd\"os-Sierpi\'nski problem finding positive integer solutions of $\sigma(n)=\sigma(n+1)$, where $\sigma(n)$ is the sum of the positive divisors of n. On the basis of that set of solutions the following empirical properties are enunciated: first, all the $\sigma(n)$, $n$ being a solution, are divisible by 6; second, the repetition of solutions leads to the formulation of a new problem: \emph{Find the natural numbers $n$ such that $\sigma(n)=\sigma(n+1)=\sigma(n+k)=\sigma(n+k+1)$ for some positive integer $k$}. A third empirical property concerns the asymptotic behavior of the function of $n$ that gives the number of solutions for $m$ less or equal to $n$, which we find to be as $n^{1/3}$. Finally some theorems related to the Erd\"os-Sierpi\'nski problem are enunciated and proved.