Cyclotomic conjecture

Proposed by Gimbert in 1999, the conjecture predicts exactly when $F_{n,k}(x)=\Phi_n(1+x+\cdots+x^k)$ is reducible over $\mathbb{Q}$, including the number of irreducible factors. In 2014, Conde, Gimbert, González, Miller, and Miret showed that it would imply nonexistence of almost Moore digraphs.

Known results

• $k=2$: proved by Lenstra Jr. and Poonen.
• $k=3$: proved by Conde et al.
• $k=4$: proved by Conde et al.; the exceptional reducible cases are $n=3$ and $n=6$.
• Computational checks cover $2<i\le 10$ and $2<k\le 200$ in related applications.

August 2026 claimed proof

Jaskaran Kaur and Hitesh Kumar claim the full factorization criterion and consequently nonexistence of almost Moore digraphs for $d>1$ and $k>2$. The arXiv manuscript is unrefereed, and no independent confirmation, gap report, or withdrawal was found.

Current status (as of August 2026): The conjecture remains unverified; the only current resolution is Kaur and Kumar’s claimed proof, whose consequence for almost Moore digraphs therefore also remains unconfirmed.