A proof of the irreducibility conjecture for Legendre polynomials

We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes in his 1890 letter to Hermite. Assuming a nontrivial factorization, we take the differences between roots belonging to distinct factors, multiply these differences, and include the leading coefficients to obtain a nonzero integer, the resultant of the two factors. We first bound the exponent with which each odd prime divides this integer, obtaining an upper bound for its odd part. We then use an identity for derivatives at roots, derived from the Legendre differential equation, to prove that the same odd part exceeds another explicit quantity. Comparing the two bounds rules out all factorizations for original degrees $n \ge 100000$. The remaining degrees are covered by Groth's previously established finite-range result.