Metadata
bibkey: huber2024a375786
authors: F. Huber
year: 2024
title: "OEIS A375786: minimum volume of a cuboid with the surface of an n-cube"
doi: null
url: https://oeis.org/A375786
claim: "Conjecture: From the integer-sided cuboids with same surface 6*n^2 always the one with the smallest edge length has the minimum volume. If there are several integer-sided cuboids having the smallest edge length, then the one with the smallest second smallest edge length has the minimum volume (checked up to a(1000))."
strata_touched:
- D5/S3/Arith/OeisA375786CuboidMinimumVolume
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A375786
Source statement
OEIS A375786, NAME (unpaginated):
a(n) is the minimum volume of an integer-sided cuboid having the same surface as a cube with edge length n.
OEIS A375786, COMMENTS (unpaginated):
Conjecture: From the integer-sided cuboids with same surface 6*n^2 always the one with the smallest edge length has the minimum volume. If there are several integer-sided cuboids having the smallest edge length, then the one with the smallest second smallest edge length has the minimum volume (checked up to a(1000)).
Mathematical scope
For n ≥ 1, list positive integer edge lengths as x ≤ y ≤ z. Equality of surfaces is xy + yz + xz = 3n². The conjecture compares the volume xyz of a least pair (x, y) in lexicographic order with every other solution, by a non-strict inequality. It does not assert uniqueness.
The repository proof uses xS ≤ 2xyz + x³ for S = xy + yz + xz, together with 3x² ≤ S. Odd n has a first-edge-one solution; even n = 2m has the solution (2, 2, 3m² − 1). The finite interval encoding bounds each edge by 3n², which follows from positivity of the other edges.
Evidence boundary
The source quotation and literature screen are supplied by preregistration issue https://github.com/the-omega-institute/trureturing/issues/18592. Completeness of the external literature search is ASSUMED-UNVERIFIED.
Computed evidence: python3 check.py 1500, exit 0, in
the-omega-institute/trureturing-experiments, commit
01e3332d967f3d17230496fd63e6800a0481d882, directory
docs/reports/oeis-a375786-cuboid-minimum-volume/.
The checker SHA-256 is
6ed9f8b113c88cfb012e371035419d3d28e382f7915e9ed8c6efbe8a852d79d3.
It verifies the conjecture through n = 1500 and matches the first
36 published terms. This computation is separate from the universal proof.
Uniqueness, the least-divisor formula, and the A375785 parity conjecture
are not asserted by the settling theorem.