Roots of unity and rational zeros of the Bernoulli-coefficient polynomials R_(k,ℓ), uniformly in ℓ
Abstract
For positive integers k and ℓ let R_(k,ℓ)(x)=Σⱼ₌₀ᵏ⁺¹(-1)^((ℓ+1)j)(tfracB₂ⱼ(2j)! tfracB₂ₖ₊₂₋₂ⱼ(2k+2-2j)!)^(ℓ)xʲ, the two-parameter family of reciprocal polynomials introduced by Maji and Sarkar [ms] and studied by Charan, Meher and Pathak [cmp], who prove that all zeros of R_(k,ℓ) except two real ones, α_(k,ℓ)>1 and 1/α_(k,ℓ), lie on the unit circle. The concluding section of [cmp] asks which zeros of R_(k,ℓ) are roots of unity, reports that computation suggests none besides ± 1 when ℓ>1, and says that "a proof of this fact seems elusive"; it also asks whether R_(k,ℓ) is irreducible, noting that this would entail the irrationality of α_(k,ℓ). For ℓ=1, where R_(k,1)(x²) is the Ramanujan polynomial, the roots-of-unity question is answered by a theorem of Murty, Smyth and Wang [msw]; for ℓ>1 nothing appears to have been known. We prove three statements, each for every ℓ ≥ 1. First, for k ∈ {8,11,13,14} no cyclotomic polynomial Φ_(d) with d ≥ 3 divides R_(k,ℓ) over ℚ: an integral model whose dependence on ℓ sits entirely in the exponents of fixed integers lets Fermat's little theorem collapse ℓ to a residue modulo p-1, so finitely many evaluations in 𝔽ₚ certify all ℓ at once, and a lower bound n ≤ 2φ(n)² for Euler's totient makes the set of candidate orders finite. Second, a 2-adic structure theorem: v₂(c_(k,j)/c_(k,0))=v₂C(k+1, j)-1 for the coefficient bases c_(k,j), so that the reduction modulo 2 of the integral model does not depend on ℓ at all; this sieve alone settles the roots-of-unity question at k ∈ {2,26,44}. Third, R_(11,ℓ) and R_(13,ℓ) have no rational zero, and every rational zero of R_(8,ℓ) and R_(14,ℓ) is ± 1; with the unit-circle theorem of [cmp] this gives the irrationality of α_(k,ℓ) for k ∈ {8,11,13,14} — the consequence of irreducibility that [cmp] names, obtained without irreducibility. The three statements are machine-checked in Lean 4; the final step to α_(k,ℓ) rests on the cited theorem of [cmp] and is not.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
RK1known2026-09-03
R_(k,l) is self-inversive for every k and l: its Bernoulli products are palindromic (c_(k,k+1-j) = c_(k,j)) and its coefficient list is palindromic up to the global sign (-1)^((l+1)(k+1))
RK2known2026-09-03
Source Proposition 4.1, both halves, for every k and l: k even and l even give R_(k,l)(1) = 0; k even and l odd give R_(k,l)(-1) = 0
RK3candidate2026-09-03
For k in 8, 11, 13, 14 and EVERY l >= 1, no cyclotomic polynomial Phi_d with d >= 3 divides R_(k,l) over Q; equivalently no zero of R_(k,l) is a root of unity other than +-1
RK4routine2026-09-03
The same statement over C: for k in 8, 11, 13, 14, every l >= 1 and every z with zⁿ = 1 for some n > 0, if R_(k,l)(z) = 0 then z = 1 or z = -1
RK5routine2026-09-03
n <= 2 * phi(n)² for every natural number n, with equality at n = 2; hence phi(d) <= N implies d <= 2N²
RK6routine2026-09-03
Negative controls: k even is needed in Proposition 4.1 (R_(1,1)(+-1)!= 0); the parity of l selects which of +-1 vanishes (R_(2,1)(1)!= 0, R_(2,2)(-1)!= 0); the restriction d >= 3 in RK3 is sharp (Phi₁ | R_(8,2), Phi₂ | R_(8,3)); the choice of k is not accidental (Phi₃ | R_(3,1), with the cofactor 3 - 13z + 3z² exhibited); and R_(8,l) has degree exactly 9 for every l
RK7measurement2026-09-03
Factorisation census: all 800 polynomials R_(k,l) with 1 <= k <= 80 and 1 <= l <= 10 factor over Q exactly as the source predicts – one irreducible factor of degree k+1 when k is odd, and (z - (-1)ˡ) times one irreducible factor of degree k when k is even – with exactly 26 exceptions, all at l = 1 and all at 3 | k
This ledger entry is reported in prose and is not bound to a Lean theorem.RK8prose2026-09-03
Two-adic structure theorem: with n = k+1, the primitive integral form of R_(k,l) reduces mod 2 to a polynomial Qₙ that does not depend on l – Qₙ = xⁿ + x^(n/2) + 1 when n is a power of two, and Qₙ = sum over 0<j<n with j subset of n bitwise of xʲ = (1+x)ⁿ + 1 + xⁿ otherwise. Hence the set of d for which Phi_d can divide R_(k,l) depends only on k; it is empty for n = 3, 27, 45, which settles the roots-of-unity question for all l at k = 2, 26, 44
This ledger entry is reported in prose and is not bound to a Lean theorem.RK9routine2026-09-03
Theorem A in Lean: v₂(c_(k,j)/c_(k,0)) = v₂(C(k+1,j)) - 1 for 0 < j < k+1, from von Staudt-Clausen; its integral-model form v₂(bⱼ) - v₂(b₀) = v₂(C(k+1,j)) - 1 for any coprime integral model; and the consequence that the mod-2 reduction of the integral model W_(B,l) is the same polynomial for every l >= 1
RK10candidate2026-09-03
For k in 2, 26, 44 and EVERY l >= 1, no cyclotomic polynomial Phi_d with d >= 3 divides R_(k,l) over Q; equivalently the only zeros of R_(k,l) that are roots of unity are +-1 – by the two-adic sieve of RK9, with no per-order and no per-l search
RK11candidate2026-09-03
For k in 11, 13 and every l >= 1, R_(k,l) has NO rational zero; for k in 8, 14 and every l >= 1, every rational zero of R_(k,l) is +-1. Hence every real zero of R_(11,l) and R_(13,l), and every real zero of R_(8,l) and R_(14,l) other than +-1, is irrational – which with the source Theorem 1.2 (alpha_(k,l) > 1) is the irrationality of alpha_(k,l) and of 1/alpha_(k,l), uniformly in l
RK12routine2026-09-03
Controls for the two-adic and rational-root layers: Theorem A model form applied at k = 2, j = 1 with its parity content; the mod-2 sieve is necessary and NOT sufficient (Phi₃ divides Q₄ mod 2, i.e. 111₂ * 111₂ = 10101₂, matching Phi₃ | R_(3,1)); l >= 1 is needed in the l-free reduction (at l = 0 the mask is 15, not 6); "no rational zero" is false for k even (R_(8,2)(1) = 0); and R_(11,1) really has a real zero (the integral model changes sign between 4 and 5)
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- For positive integers k, ℓ, Maji and Sarkar (arXiv:2306.10283) introduced
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7