Let M(n) denote the minimum number of distinct positive integers in an n-tuple whose second elementary symmetric sum equals its nth elementary symmetric sum. For n∈N_≥ 3, consider the values of n for which M(n)=2. Conjecture. The only values of n∈N_≥ 3 such that M(n)=2 are n=3,4,5,11,41156. In particular, liminf_n rightarrow+∞M(n)=3. This is motivated by the reduction of the case M(n)=2 to integral points on the curves y^2=8x^k+4kx^2-4kx+1, together with explicit analyses for small k and the fact that the associated curves have genus at least 2 for k≥ 5. The asserted absence of further integral points for larger k is based on numerical calculations and remains open.

The conjecture on the values of n for which M(n)=2

From papers

Let $M(n)$ denote the minimum number of distinct positive integers in an $n$ -tuple whose second elementary symmetric sum equals its $n$ th elementary symmetric sum. For $n\in\mathbb{N}_{\geq 3}$ , consider the values of $n$ for which $M(n)=2$ .

Conjecture. The only values of $n\in\mathbb{N}_{\geq 3}$ such that $M(n)=2$ are

n=3,4,5,11,41156.

In particular,

\liminf_{n\rightarrow+\infty}M(n)=3.

This is motivated by the reduction of the case $M(n)=2$ to integral points on the curves $y^2=8x^k+4kx^2-4kx+1$ , together with explicit analyses for small $k$ and the fact that the associated curves have genus at least $2$ for $k\geq 5$ . The asserted absence of further integral points for larger $k$ is based on numerical calculations and remains open.

Progress summary

Open

The proposed list remains an unproved conjecture, with no verified new examples or proof found.

The conjecture claims that exactly $n=3,4,5,11,41156$ satisfy $M(n)=2$ , and therefore that the minimum number of distinct entries eventually has lower limit $3$ . It was stated in a 2022 paper, where the final exclusions were explicitly left conjectural.

Known results

The case $M(n)=2$ reduces to integral points on $y^2=8x^k+4kx^2-4kx+1$ .
Explicit analysis for $k=3$ and $k=4$ yields $n=3,4,5,11,41156$ .
For $k\geq5$ , the associated curves have genus $g\geq2$ ; numerical calculations suggest no further relevant integral points, but do not prove this.
A 2026 paper treats broader elementary-symmetric equations but reports no proof or counterexample for this specific conjecture.

Current status (as of August 2026): The reduction and the explicitly analyzed cases are established, but completeness of the list and the resulting claim about $\liminf_{n\to+\infty}M(n)$ remain open.

Sources

Sources & referencesView supporting material

Primary source

Piotr Miska and Maciej Ulas, “On the Diophantine equation σ_2(X_n)=σ_n(X_n)”, arXiv:2203.03942 (2022).

1 week ago

The proposed classification

M(n)=2\quad\Longleftrightarrow\quad n\in\{3,4,5,11,41156\}

fails in both directions.

First, take the nondecreasing positive integer tuple

X=(\underbrace{1,\ldots,1}_{43},7,7,7,7).

Its product is

\sigma_{47}(X)=7^4=2401,

and separating pairs according to how many entries equal $7$ gives

\begin{aligned} \sigma_2(X) &=\binom{43}{2}+43\cdot4\cdot7+\binom42\,7^2\\ &=903+1204+294\\ &=2401. \end{aligned}

Thus $X$ is a solution with exactly two distinct entries, so $M(47)\le2$ . A constant solution of length $47$ would require

\binom{47}{2}t^2=t^{47}, \qquad\text{hence}\qquad t^{45}=1081=23\cdot47.

No positive integer satisfies this equation: $t=1$ gives $1$ , while $t\ge2$ gives $t^{45}\ge2^{45}>1081$ . Consequently

M(47)=2,

although $47$ is absent from the proposed list.

Second, the constant tuple $(3,3,3)$ satisfies

\sigma_2(3,3,3)=3\cdot3^2=27=3^3=\sigma_3(3,3,3),

and therefore

M(3)=1\ne2,

although $3$ appears in the proposed list.

Both underlying examples already occur immediately before the conjecture in its source: the displayed conjectural classification is inconsistent with those preceding results. A corrected completeness statement would be a different problem.

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Shivam Patel · 23 hours ago