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math.NT — Number Theory

A 256-step certificate for Turán's pure power-sum constant

Contributed by Ricardo Pinho

We give an exact-input certificate for the upper bound C_42 <= 0.6888863712 in the pure power-sum problem. The construction refines the step profiles of Griego, Röhrig and Plata to 256 middle blocks, without changing their coefficient-cancellation mechanism. It improves the strongest public proposal located in our 22 October 2026 audit, 0.688970, by 0.0000836288. An unmodified public interval checker and a separately written Arb checker verify the strict finite inequalities. The asymptotic implication is proved in the note; no explicit finite threshold, optimality, human peer review or formal verification is claimed.

Primarily AI-generated textHuman understanding: some partsTurán power sumscoefficient cancellationcomputer-assisted proofinterval arithmeticrigorous numerics

Oct 23, 2026PDF ↗View work →

hexagon:2610.00463v1

math.NT — Number Theory

Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

Contributed by K. MAHESH KRISHNA

Motivated by the Smale mean value and the Dubinin–Sugawa dual mean value theorems, we formulate non-Archimedean Smale mean value and Dubinin–Sugawa dual mean value problems. We show that if a universal constant satisfying the non-Archimedean Smale mean value problem exists, it must be greater than or equal to one and if a universal constant satisfying the non-Archimedean Dubinin–Sugawa dual mean value problem exists, it must be less than or equal to one. We also derive other bounds whenever roots of unity exist in a non-Archimedean field.

Primarily human-written textHuman understanding: all partsDubinin-Sugawa dual mean value theoremNon-Archimedean field.Smale mean value theorem

Oct 22, 2026PDF ↗View work →

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A 256-step certificate for Turán's pure power-sum constant

Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

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