Let n,m∈Z and A,B∈Zsetminus0 satisfy 1≤ m<n and gcd(n,mB)=1. Suppose that f(X)=X^n+A(BX+1)^m∈Z[X] is irreducible, let alpha be a root of f, and set K=Q(alpha). Then f is monogenic, meaning O_K=Z[alpha], if and only if both A and n^n+(-1)^n+mB^n(n-m)^n-mm^mA are square-free integers.

Lenny Jones’ conjecture

Let $n,m\in\mathbb{Z}$ and $A,B\in\mathbb{Z}\setminus\{0\}$ satisfy $1\le m<n$ and $\gcd(n,mB)=1$ . Suppose that $f(X)=X^n+A(BX+1)^m\in\mathbb{Z}[X]$ is irreducible, let $\alpha$ be a root of $f$ , and set $K=\mathbb{Q}(\alpha)$ . Then $f$ is monogenic, meaning $\mathcal{O}_K=\mathbb{Z}[\alpha]$ , if and only if both $A$ and $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ are square-free integers.

References

Primary source

arXiv

Additional references

A Local Approach to Monogenity with an Application to Lenny Jones' Conjecture — arXiv — Michail Karatarakis, Sumandeep Kaur

algebraic-number-theory monogenic-polynomials

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to settle the conjecture for its stated polynomial family, but the claim has not been independently checked.

Jones proposed the conjecture in 2021 concerning monogenicity in a specified family of integer polynomials. Earlier work reported the conjecture true for the family covered by its stated conditions.

Known results

A paper published on November 21, 2023, states an if-and-only-if square-freeness criterion involving $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ .
A 2026 paper partially proves a related Jones conjecture for reciprocal polynomials and obtains new monogenic examples in degrees $5$ and $10$ .

September 2026 claimed proof

Michail Karatarakis and Sumandeep Kaur’s preprint gives a local criterion: monogenicity is characterized by square-freeness of two explicit integers, with infinite non-monogenic families. It claims the conjecture for its specified polynomial family, but no independent mathematical assessment was found.

Current status (as of September 2026): The conjecture is claimed solved for the stated family, while the new proof remains unverified.

Sources

No solutions have been posted yet.