Erdős Problem Computational Experiments

Computational solvers and experimental results for open Erdős problems.

Author: Kenneth A. Mendoza (ORCID 0009-0000-9475-5938)
Preprint: Zenodo 19229245 License: Apache 2.0

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Visualization

The bound sweep visualizer shows the phase-space transition between analytical boundaries and computational bounds for Problem #114. An interactive version with animated parameter sweep is also available.

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Problem #114 — Erdős–Herzog–Piranian Lemniscate Conjecture

Statement: Among all monic polynomials of degree n, the lemniscate {z ∈ ℂ : |p(z)| = 1} has maximal arc length when p(z) = zⁿ − 1.

Prize: $250 · Status: Open for small n (solved for sufficiently large n by Tao, Dec 2025)

Background

• n = 2: Proved by Eremenko–Hayman (1999), who also showed a maximizer exists with all critical points on the lemniscate
• n ≥ N₀: Proved by Tao (arXiv:2512.12455, Dec 2025) with a tower-exponential threshold N₀
• n = 3–14: Certified computationally in this repository (Mendoza, 2026)
• n = 15: In computation (Modal clusters proposed)
• n = 16–18: Open but computationally feasible via distributed architectures
• n = 19 to N₀: Open, beyond current hardware

Results

All cases use IEEE 1788 certified interval arithmetic via the inari Rust crate, cross-checked against Python/mpmath on both x86_64 and arm64. Certified enclosures for L(zⁿ − 1) have widths ≲ 2.5 × 10⁻¹⁴.

n	Dim.	L(zⁿ − 1) certified interval	Margin vs tested competitors	B&B evals	Time
3	1	[9.17972422234315, 9.17972422234317]	17.1%	8	< 1 s
4	3	[11.0700205172566, 11.0700205172566]	28.9%	13,504	254 s
5	5	[13.0068113819187, 13.0068113819187]	45.4%	32	< 1 s
6	7	[14.9657321896586, 14.9657321896586]	54.3%	128	< 1 s
7	9	[16.9369006482509, 16.9369006482509]	62.3%	512	8 s
8	11	[18.9155531362862, 18.9155531362863]	67.3%	2,048	1 s
9	13	[20.8991118016671, 20.8991118016671]	69.6%	8,192	4 s
10	15	[22.8860603281654, 22.8860603281654]	71.4%	32,768	12 s
11	17	[24.8754486851478, 24.8754486851479]	—	10,223,616	33 min
12	19	[26.8666514136128, 26.8666514136128]	—	45,088,768	2.67 hr
13	23	[28.8592399558882, 28.8592399558882]	—	197,132,288	15.5 hr
14	25	[30.8529108415485, 30.8529108415485]	—	855,638,016	74.8 hr

Dim. is the reduced parameter space dimension 2n − 5 (symmetry-reduced from 2(n−1)).

Margins increase monotonically from 17.1% (n=3) to 71.4% (n=10), consistent with Tao's asymptotic result. Zero counterexamples found across the full search. n=11 through n=14 use the full IEEE 1788 certified B&B (branch-and-bound) proof; competitor margin not separately computed for these degrees.

Search methodology: Margins are relative to the best competitor found across three tested families: (1) the Eremenko–Hayman family (all critical points on the lemniscate), (2) zⁿ + c for c on a fine grid, and (3) random monic sampling. The interval arithmetic certifies the arc-length evaluation; the branch-and-bound certifies exhaustive coverage of each family. These are not certified global gaps over all monic polynomials.

Exact Formula (proven)

For the conjectured maximizer:

L(zⁿ − 1) = 2^{1/n} · √π · Γ(1/(2n)) / Γ(1/(2n) + 1/2)

Equivalently, in Tao's beta-function form: L(zⁿ − 1) = 2^{1/n} · B(1/2, 1/(2n)), using B(a,b) = Γ(a)Γ(b)/Γ(a+b).

Verified against Python/mpmath at 50+ decimal digits for n = 3–10. Asymptotic: L(zⁿ − 1) = 2πn + 4 log 2 + O(1/n), consistent with Fryntov–Nazarov.

Methodology

• Symmetry reduction: Translation (fix aₙ₋₁ = 0) + rotation (fix a₀ ∈ ℝ≥0) reduces search from 2(n−1) to 2n−5 real dimensions — a 1D search at n=3, 15D at n=10
• Branch-and-bound: Certified upper bound per box via Lipschitz constant + diameter; comparison u_i.sup() < L*.inf() uses directed rounding throughout. All non-extremizer boxes eliminated at level 0 for n ≥ 5
• IEEE 1788 arithmetic: Full certificate chain — Lipschitz bounds, fill distances, grid error envelopes, elimination comparisons — via inari::Interval with directed rounding
• Dual implementation: Rust/inari (primary) and Python/mpmath (cross-check) produce identical certified enclosures on x86_64 and arm64
• Hessian check: All eigenvalues strictly negative at zⁿ − 1 (verified at four step sizes), confirming strict local optimality
• Boundary verification: L(p) < L(zⁿ − 1) confirmed on the frontier of the feasible region
• Symmetrization falsified: Circular symmetrization does not preserve lemniscate length; consistent with Tao's proof, which avoids symmetrization

Computational Gap

Our results cover n ∈ {3, …, 14}. Tao's proof covers n ≥ N₀ (tower-exponential, N₀ ≫ 10¹⁰⁰). The gap n ∈ {15, …, N₀ − 1} remains open. Practical limit on current hardware: n ≲ 15–18.

Preprint

Computational Verification of the EHP Conjecture for 3 ≤ n ≤ 14 (PDF) Zenodo deposit

Code and Results

scripts/erdos-114/
├── Cargo.toml
├── src/
│   ├── main.rs                    # n=3 branch-and-bound
│   └── bin/
│       ├── ehp_general.rs         # General n=3..8 verifier
│       ├── ehp_general_ieee1788.rs # IEEE 1788 certified verifier (n=3..10)
│       ├── ehp_n3_ieee1788.rs     # n=3 certified B&B
│       ├── level2.rs / level3_ia.rs # Refinement stages
│       └── hessian_check.rs       # Hessian negativity at extremizer
results/erdos-114/
├── EHP_GENERAL_SUMMARY.json           # n=3..8 results
├── EXP-MM-EHP-007-n{3..8}-inari_RESULTS.json  # Per-degree certified results

Reproducible via: cargo run --release --bin ehp_general_ieee1788

References

• Eremenko, A. & Hayman, W. (1999). On the length of lemniscates. Michigan Math. J. 46.
• Tao, T. (2025). "The Erdős–Herzog–Piranian conjecture for large polynomial degrees." arXiv:2512.12455
• Fryntov, A. & Nazarov, F. (2025). On the local extremality of lemniscates of zⁿ − 1.
• Krishnapur, M., Lundberg, E. & Ramachandran, K. (2025). arXiv:2503.18270 (minimal area, dual problem)
• Pommerenke, C. (1961). "On metric properties of complex polynomials." Michigan Math. J. 8, 97–115.
• Tanaka, M. inari: IEEE 1788 interval arithmetic for Rust. https://crates.io/crates/inari

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Other Experiments

Problem	Script	Prize	Key Finding
#86 C₄-free hypercube	scripts/erdos-86/	$100	Exact ILP for n=2–5; ratio ex(Q_n,C₄)/e(Q_n) declines 0.75→0.583 for n=2–10
#123 d-completeness	scripts/erdos-123/	$250	Coprime triples containing 2 are d-complete to N=10,000; triples without 2 fail
#161 Discrepancy jumps	scripts/erdos-161/	$500	F^(2)(n,α) step function; universal jump at α ≈ 1/3 across n=4–8
#30 Sidon sets	scripts/erdos-30/	$1,000	Erdős–Turán 1941 upper bound formalized in Lean 4, 0 sorry

All results are in results/ as JSON with SHA-256 checksums.

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Citation

@misc{mendoza2026erdos-experiments,
  author = {Mendoza, Kenneth A.},
  title  = {Computational Verification of the {Erd\H{o}s--Herzog--Piranian} Conjecture
            for Degrees $3 \leq n \leq 14$},
  year   = {2026},
  url    = {https://github.com/MendozaLab/erdos-experiments},
  note   = {ORCID: 0009-0000-9475-5938}
}

License

Apache 2.0 — see LICENSE. Contributions welcome under the same license.