A proof of the irreducibility conjecture for Legendre polynomials
Abstract
We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer j≥ 1, both P2j(x) and P2j+1(x)/x are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from Pn when present, multiply by a suitable constant, and express the resulting polynomial as a product of two integral polynomials A(x2) and B(x2). We then construct the resultant and its odd part, R=Rest(A,B), ℛ=|R|/2v2(R). Put k=deg A≤deg B and m=⌊ n/2⌋. Using the classical auxiliary polynomial Un and the differential identity (1−x2)(Pn'Un−PnUn')=1−Pn2, together with orthogonality, divisibility properties of the coefficients, and a least-common-multiple estimate, we obtain an upper bound for ℛ. On the other hand, the Legendre differential equation gives a lower bound for the absolute value of the derivative of AB at each root of A. We use Chebyshev polynomials and Hadamard's inequality to bound the product of the squared pairwise differences of these roots, and combine this with estimates for the leading coefficients and the power of $2$ in R to obtain a lower bound for ℛ. These estimates yield mlog4−log(4(m+1)2)/√m <logℛ/k <(m+√(2n−1))log3 (n≥64). For every n≥256, the lower bound strictly exceeds the upper bound, giving a contradiction. Combining this with established irreducibility results and explicit integer comparisons for the remaining degrees proves irreducibility in every degree.
Related
- A Proof of the Common Root Conjecture for Legendre Polynomials
- The Odlyzko-Poonen Conjecture on Irreducibility of Random Polynomials
- A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant
- Proof of Almkvist's conjecture on the unimodality of partition polynomials
- Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture
BibTeX
@misc{deng2026proof,
title = {{A proof of the irreducibility conjecture for Legendre polynomials}},
author = {Zikang Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22336},
eprint = {2609.22336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22336},
}
RIS
TY - UNPB
TI - A proof of the irreducibility conjecture for Legendre polynomials
AU - Zikang Deng
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.22336
UR - https://arxiv.org/abs/2609.22336
AB - We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer j≥ 1, both P_2j(x) and P_2j+1(x)/x are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from P_n when present, multiply by a suitable constant, and express the resulting polynomial as a product of two integral polynomials A(x^2) and B(x^2). We then construct the resultant and its odd part, R=Res_t(A,B), ℛ=|R|/2^v_2(R). Put k=deg A≤deg B and m=⌊ n/2⌋. Using the classical auxiliary polynomial U_n and the differential identity (1−x^2)(P_n'U_n−P_nU_n')=1−P_n^2, together with orthogonality, divisibility properties of the coefficients, and a least-common-multiple estimate, we obtain an upper bound for ℛ. On the other hand, the Legendre differential equation gives a lower bound for the absolute value of the derivative of AB at each root of A. We use Chebyshev polynomials and Hadamard's inequality to bound the product of the squared pairwise differences of these roots, and combine this with estimates for the leading coefficients and the power of $2$ in R to obtain a lower bound for ℛ. These estimates yield mlog4−log(4(m+1)^2)/√m <logℛ/k <(m+√(2n−1))log3 (n≥64). For every n≥256, the lower bound strictly exceeds the upper bound, giving a contradiction. Combining this with established irreducibility results and explicit integer comparisons for the remaining degrees proves irreducibility in every degree.
ER -