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

About 4 years old · traced to 2022

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The stated classification is contradicted by examples in its source and by a posted calculation, but no complete corrected classification is known.

Miska and Ulas (2022) formulate the conjecture that the listed values are exactly those with $M(n)=2$ , and deduce the claimed lower-limit consequence. Their paper also contains computations that conflict with this formulation.

Known results

The two-value case reduces to integral points on $y^2=8x^k+4kx^2-4kx+1$ .
For $k=3$ , the relevant points yield $n=3$ and $n=11$ ; the source also states $M(3)=1$ .
For $k=4$ , the points $(x,y)=(7,141)$ and $(172,83679)$ yield $n=47$ and $n=41156$ .
For $k\geq5$ , the curves have genus at least $2$ ; finiteness follows from Faltings, but excluding all further relevant integral points remains numerical and conjectural.

Posted attempt

A posted calculation claims a two-value $47$ -tuple with forty-three entries equal to $1$ and four equal to $7$ , proving $M(47)=2$ , while the constant tuple $(3,3,3)$ gives $M(3)=1$ . The attempt has not been independently verified, although these discrepancies are consistent with the source’s displayed computations.

Current status (as of August 2026): The proposed conjecture is false as stated if the displayed $n=47$ example is valid, while a complete corrected classification and the resulting claim about $\liminf_{n\to\infty}M(n)$ remain open.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.