Checked 2026-10-08. This is a prior-art assessment, not a proof audit or a priority certificate. No producer proof or computation was changed.
Exact comparison target
Assume every nontrivial zero of the Riemann zeta-function has real part at most 7/8, equivalently there are no such zeros in the open half-plane Re(s)>7/8. For every integer n>=1, prove a prime exists strictly between n^7 and (n+1)^7. Boundary zeros with real part exactly 7/8 are permitted. Full RH is a stronger hypothesis; eventual existence, finite verification, and almost-prime existence are different conclusions.
Assessment: adjacent prior results authenticated; priority unresolved. This bounded search did not authenticate an earlier statement with this exact hypothesis and all-n conclusion. That absence is not evidence of priority. The useful distinction in the present artifact is explicit coverage of every n under the stated 7/8 assumption, not discovering eventual seventh-power intervals or the use of zero-density estimates.
Closest primary sources inspected
| Primary source and precise location | Actual result and comparison |
|---|---|
| Ethan S. Lee, Minimal zero-free regions for results on primes between consecutive perfect kth powers, Acta Arithmetica, online 1 October 2026, DOI 10.4064/aa260304-20-7; Theorems 1.2, 1.4 and Proposition 3.1. Publisher full text (https://www.impan.pl/shop/en/publication/transaction/download/product/116568); arXiv v2 (https://arxiv.org/pdf/2602.14340v2). | Theorem 1.2 proves all-n 86th powers unconditionally. Theorem 1.4 proves all-n k in {85,80,75,70} from specified finite-height Littlewood-shaped zero-free regions. These hypotheses are not an assertion of full RH. Proposition 3.1 explicitly requires k>=11, so it cannot simply be specialized to k=7. This is close prior art for improving explicit all-n powers through zero-free information. |
| Michaela Cully-Hugill and Daniel R. Johnston, On the error term in the explicit formula of Riemann–von Mangoldt II, Theorem 1.4 and Section 5. Primary preprint v3 (https://arxiv.org/html/2402.04272v3). | Unconditional all-n 90th powers. Explicit formulas combined with density estimates are established technology, not a novel general mechanism of the present proof. The theorem inspected does not give all-n seventh powers under the 7/8 assumption. |
| Marc Chamberland and Armin Straub, Weakening the Legendre Conjecture, arXiv:2602.22502v1, Theorem 1.2 and its proof in Section 3. Primary full text (https://arxiv.org/html/2602.22502v1). | Under full RH, every real x>=1 has a prime between x^(2+delta) and (x+1)^(2+delta) when delta>=1/4. Thus seventh powers for all integer n are already known under this stronger assumption. Integer seventh-power endpoints cannot themselves be prime, so closed endpoints do not obstruct comparison with our strict intervals. This does not imply the same result from the weaker 7/8 assumption. |
| Michaela Cully-Hugill, Primes between consecutive powers, J. Number Theory 247 (2023), 100–117; Theorem 1 and Section 4, printed p.14. Primary preprint v3 (https://arxiv.org/pdf/2107.14468v3). | Unconditional all-n 155th powers, and primes between consecutive cubes for n>=exp(exp(32.892)). The latter is an explicit eventual result, not all-n cubes. This is already much stronger in the asymptotic exponent than eventual seventh powers. |
| Adrian Dudek, An Explicit Result for Primes Between Cubes, Theorem 1.1. Primary preprint (https://arxiv.org/pdf/1401.4233). | Unconditional primes between consecutive cubes for n>=exp(exp(33.217)) in the inspected version. It supplies an older, direct primary antecedent for eventual small powers. Its zero-density and explicit-formula method also predates this work. |
| Daniel R. Johnston, Jonathan P. Sorenson, Simon N. Thomas and Jonathan E. Webster, Primes and almost primes between cubes, Theorems 1.3–1.4. Primary full text (https://arxiv.org/html/2601.15564). | Theorem 1.3 verifies prime-containing cube intervals in a finite range, n^3<=1.649 times 10^40. Theorem 1.4 gives an integer with at most two prime factors in every cube interval. Neither is an all-n theorem for actual primes between cubes. |
The elementary implication from eventual cubes to eventual seventh powers does not require a new analytic theorem: for sufficiently large integer n, set m=ceil(n^(7/3)). Then m^3>=n^7 and m+1<n^(7/3)+2<(n+1)^(7/3), so the cube interval (m^3,(m+1)^3) lies in the seventh-power interval. The finite range left by such enormous cube thresholds is precisely why an all-n result is materially different.
The current multistrip construction uses published density constants rather than claiming new zero-density theory. See also Kadiri–Lumley–Ng, primary paper (https://arxiv.org/pdf/2101.12263), Lemma 4.14, equation (4.71), and Table 1. Their exact applicability and the explicit-formula source were checked separately in the source audit and the seventh-power audit.
Search scope and safe wording
Searches included combinations of “seventh powers”, “primes between”, “7/8”, “seven-eighths”, “quasi-Riemann hypothesis”, “short intervals”, and “consecutive powers”; follow-up searches used author names and exact titles. Sources searched were the general web index, arXiv full texts, publisher full texts, and the reference chains of the papers above. Several general queries returned poorly matched results. This was not a systematic MathSciNet/zbMATH search, exhaustive forward-citation analysis, or a search of every recent OpenAI-consequence manuscript. Some current arXiv endpoints are mutable; explicit versions are supplied where verified.
Suggested description: “Assuming that zeta has no nontrivial zeros with real part greater than 7/8, we give an explicit proof, with independently replayed interval arithmetic, of primes between consecutive seventh powers for every positive integer. This improves this project's earlier ninth-power consequence. Related all-n results under full RH and unconditional eventual results for smaller powers are classical or already published; priority for this exact weaker-hypothesis statement is not established here.”
Do not describe the result as unconditional, the first all-n seventh-power theorem, a new asymptotic prime-gap exponent, or a new general density method. The local written proof and arithmetic audit are distinct from Lean formalization.