Let f=a_0+a_1x+…+a_nx^n∈ Z[x], where a_0=pm p^k for a prime p and positive integer k. Suppose that a_1=a_2=…=a_q-1=0 for a prime q≤ n. Koley and Reddy's conjecture. If p nmid a_qa_n, q nmid k, and p^k>|a_q|+|a_q+1|+…+|a_n|, then f is irreducible over Q. The paper states that this conjecture follows from an earlier theorem, so it is solved rather than open.

Koley and Reddy's irreducibility conjecture for integer polynomials

Koley and Reddy’s conjecture asserts irreducibility under a prime-power constant term, a prescribed initial gap in the coefficients, coprimality conditions, and a coefficient-size inequality. A 2023 preprint states that the conjecture follows from a stronger irreducibility theorem proved using Newton polygons.

Known results

A 2020 paper records related irreducibility criteria for sparse integer polynomials with constant term $\pm p^u$ .
The earlier catalogue source described the conjecture as open in October 2023, before the relevant proof was identified.

2023 stronger-theorem proof

The preprint proves irreducibility when $p\nmid a_m$ , $\gcd(u,m)=1$ , and $p^u>|a_n|+\cdots+|a_m|$ ; choosing $m=q$ and using that $q$ is prime yields the Koley–Reddy statement. Its argument combines Newton polygons with the coefficient-size bound. No retrieved source reports a counterexample, withdrawal, or correction.

Current status (as of September 2026): The conjecture is claimed settled by the 2023 preprint’s stronger theorem, while independent verification of that proof is not recorded here.

Koley and Reddy's irreducibility conjecture for integer polynomials

Primary source

Progress summary

Solutions 0