The denominator 19 is forced: the conditional constant 1/2+1/3+tfrac219 for integer polynomials with minimal integrals
Abstract
For a positive integer N put ℓ_N=lcmop(1,2,…,N) and let S_N={P ∈ ℤ[x]:deg P<N, ∫₀¹P(x) dx=1/ℓ_N} be D. Bazzanella's set of integer polynomials with the smallest possible positive integral on [0,1]; it is the set on which the Gelfond–Schnirelman–Nair elementary lower bound for psi is an equality. Sanna asked how large δ_Q=liminf_Ndeg(Q_N)/N can be over sequences of explicit polynomials Q_N each dividing some member of S_N, and Bazzanella and Sanna have recently observed that Q_N=x^(⌊ N/2⌋)(1-x)^(⌊ N/3⌋)(1-x-x²)^(⌊ N/19⌋) would give δ_Q=1/2+1/3+frac219=(107)/(114)=0.9385964912…, verifying the required divisibility by a coefficient search for N ≤ 1000. We prove a Bézout criterion, in both directions, for the whole three-factor family xᵃ(1-x)ᵇ(1-x-x²)ᶜ — divisibility of some member of S_N is equivalent to a gcd of N-a-b-2c explicit integers being 1 — and use it to settle four things. The divisibility holds for every N ≤ 2000, twice the published range. The denominator 19 is the least one that can occur: for 13 ≤ d ≤ 18 the analogous sequence fails already at N=13,15,15,18,18,18, so no member of that one-parameter family can produce a constant larger than 107/114. Each of the three exponents is maximal. Finally, and unconditionally in N: no prime p with deg Q<p ≤ N<p² divides the integral ℓ_N∫₀¹Q(x)x^(p-1-deg Q)dx, so an obstructing prime p ≤ N is at most deg Q_N ≈ 0.9386 N. Every theorem below is machine-checked in Lean 4; S[sec:verif] says exactly what is checked and on what computation each claim rests.
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Claim ledger
Stated results
MI1known data2026-09-03
For every 1 <= N <= 200 there is a polynomial P with deg P < N, integral₀¹ P = 1/lcm(1..N) (i.e. P in S_N) and Q_N | P in Z[x], where Q_N = x^(floor(N/2))(1-x)^(floor(N/3))(1-x-x²)^(floor(N/19)). Kernel decide +kernel, no axioms at all.
MI2known data2026-09-03
The same statement for every 1 <= N <= 1000 – exactly the source's own window – by native_decide (93 s against its Mathematica FindInstance loop).
MI3candidate2026-09-03
The same statement for every 1001 <= N <= 2000, beyond the source's verified window; so the property of (1.5) holds on a window twice as long as the one the constant 0.938... is conditional on.
MI4routine2026-09-03
The criterion, both directions: for Q = xᵃ (1-x)ᵇ (1-x-x²)ᶜ with deg Q < N, some P in S_N is divisible by Q if and only if gcd b(i): 0 <= i <= N-1-deg Q = 1, where b(i) = lcm(1..N) * integral₀¹ Q(x) xⁱ dx is an integer. With the coefficient functional identified with the interval integral over [0,1].
MI5candidate2026-09-03
d = 19 is the least denominator for which x^(floor(N/2))(1-x)^(floor(N/3))(1-x-x²)^(floor(N/d)) has both delta = 1/2+1/3+2/d < 1 and the property: d = 13,...,18 are refuted at N = 13, 15, 15, 18, 18, 18 respectively (three exponent triples (6,4,1), (7,5,1), (9,6,1)), and no P in S_N is divisible by those polynomials at all. Hence 0.9385964912... is the largest value the source's own one-parameter family can produce.
MI6candidate2026-09-03
Every exponent of Q_N is maximal: no P in S₅ is divisible by x³(1-x) or by x²(1-x)², and no P in S₁3 is divisible by x⁶(1-x)⁴(1-x-x²) – i.e. raising any one of the three exponents of Q_N by 1 breaks the property, at N = 5, 5, 13. With positive controls at the same N (Q₅ and Q₁8 do divide members of S₅, S₁8).
MI7candidate2026-09-03
Unconditional, uniform in N: for Q = xᵃ(1-x)ᵇ(1-x-x²)ᶜ and any prime p with deg Q < p <= N < p², the index i = p-1-deg Q has p not dividing b(i). Hence every prime factor of gcdb(i) is at most deg Q; for the source's Q_N that bound is floor(N/2)+floor(N/3)+2 floor(N/19), so an obstruction to (1.5) can only sit at a prime below about 0.9386 N.
MI8measurement2026-09-03
The property of (1.5) holds for every N <= 10000 – ten times the source's window – and also at N = 12000, 20000, 30000; with the measured cost curve, and N >= 5*10⁴ parked at an extrapolated 9.5 CPU-h for a single value and 56 CPU-h for a full sweep to N <= 20000.
This ledger entry is reported in prose and is not bound to a Lean theorem.MI9measurement2026-09-03
Barrier for the shape: 1,313,012 exponent rules (floor(alpha N), floor(beta N), floor(gamma N)) on the 1/570 grid with alpha, beta <= 1/2, alpha+beta <= 5/6 and delta = alpha+beta+2 gamma > 0.9385964912 all fail at some N <= 400; and no fourth factor drawn from a 15-polynomial library, at any denominator d <= 60, survives N <= 160. But per-N maxima are far higher: maxa+b+2c: xᵃ(1-x)ᵇ(1-x-x²)ᶜ feasible for N / N reaches 0.971 at N = 70.
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- For a positive integer N write ℓ_N = lcm1,…,N and I(P) = ∫₀¹ P(x) dx. D. Bazzanella's set
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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