Exceptions to the Erd\H os--Straus--Schinzel conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908768379338752 |
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| author | Pomerance, Carl Weingartner, Andreas |
| author_facet | Pomerance, Carl Weingartner, Andreas |
| contents | A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpiński, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exceptions to the Erd\H os--Straus--Schinzel conjecture Pomerance, Carl Weingartner, Andreas Number Theory 11D68, 11D72, 11N37 A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpiński, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions. |
| title | Exceptions to the Erd\H os--Straus--Schinzel conjecture |
| topic | Number Theory 11D68, 11D72, 11N37 |
| url | https://arxiv.org/abs/2511.16817 |