Huber’s Minimum-Volume Cuboid Conjecture

Abstract

The smallest first edge, then the smallest second edge, gives minimum cuboid volume.

Definition 1.1 (Cuboids of a prescribed surface).

$$\forall n\in \mathbb{N},cuboids\left(n\right)=\mathrm{Finset}.\mathrm{filter}\left((\mathrm{fun}p:(\mathbb{N}\times (\mathbb{N}\times \mathbb{N}))\mapsto ((p.1\le p.2.1)\wedge ((p.2.1\le p.2.2)\wedge (p.1\cdot p.2.1+p.2.1\cdot p.2.2+p.1\cdot p.2.2=3\cdot n^2)))),\mathrm{Finset}.\mathrm{product}\left(\mathrm{Finset}.\mathrm{Icc}\left(1,3\cdot n^2\right),\mathrm{Finset}.\mathrm{product}\left(\mathrm{Finset}.\mathrm{Icc}\left(1,3\cdot n^2\right),\mathrm{Finset}.\mathrm{Icc}\left(1,3\cdot n^2\right)\right)\right)\right)$$

Formalization. D5/S3/Arith/OeisA375786CuboidMinimumVolume.cuboids (✓ std3).

Citation. F. Huber (2024). OEIS A375786: minimum volume of a cuboid with the surface of an n-cube. URL: https://oeis.org/A375786.

Commentary.

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.” The triple p represents edges x = p.1, y = p.2.1 and z = p.2.2, ordered x ≤ y ≤ z. The equation xy + yz + xz = 3n² says the surface is 6n². Each positive edge is at most 3n², so the three inclusive intervals contain every such cuboid. Finset.product forms the right-associated product, and Finset.filter keeps the ordered solutions.

Definition 1.2 (Volume).

$$\forall p\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),volume\left(p\right)=p.1\cdot p.2.1\cdot p.2.2$$

Formalization. D5/S3/Arith/OeisA375786CuboidMinimumVolume.volume (✓ std3).

Citation. F. Huber (2024). OEIS A375786: minimum volume of a cuboid with the surface of an n-cube. URL: https://oeis.org/A375786.

Commentary.

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.” The volume of the triple is the product of its three edge lengths.

Definition 1.3 (Huber’s conjecture).

$$claim\iff (\forall n\in \mathbb{N},(1\le n)\Rightarrow (\forall p\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(p\in cuboids\left(n\right))\Rightarrow ((\forall q\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(q\in cuboids\left(n\right))\Rightarrow ((p.1<q.1)\vee ((p.1=q.1)\wedge (p.2.1\le q.2.1))))\Rightarrow (\forall q\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(q\in cuboids\left(n\right))\Rightarrow (volume\left(p\right)\le volume\left(q\right))))))$$

Formalization. D5/S3/Arith/OeisA375786CuboidMinimumVolume.claim (✓ std3).

Citation. F. Huber (2024). OEIS A375786: minimum volume of a cuboid with the surface of an n-cube. URL: https://oeis.org/A375786.

Commentary.

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)).” Here n is any natural number at least one. The hypothesis on p chooses the least pair (p.1, p.2.1) in lexicographic order. The conclusion compares its volume with every member of the same finite set, using a non-strict inequality.

Theorem 1.4 (The lexicographic minimum has minimum volume).

$$\forall n\in \mathbb{N},(1\le n)\Rightarrow (\forall p\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(p\in cuboids\left(n\right))\Rightarrow ((\forall q\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(q\in cuboids\left(n\right))\Rightarrow ((p.1<q.1)\vee ((p.1=q.1)\wedge (p.2.1\le q.2.1))))\Rightarrow (\forall q\in (\mathbb{N}\times (\mathbb{N}\times \mathbb{N})),(q\in cuboids\left(n\right))\Rightarrow (volume\left(p\right)\le volume\left(q\right)))))$$

Proof. Machine-checked in Lean as D5/S3/Arith/OeisA375786CuboidMinimumVolume.result (✓ std3). ∎

Resolves. Problems/oeis-a375786-cuboid-minimum-volume-huber (proved) by D5/S3/Arith/OeisA375786CuboidMinimumVolume.result.

Source. Repository-derived.

Acknowledgement. F. Huber (2024). OEIS A375786: minimum volume of a cuboid with the surface of an n-cube. URL: https://oeis.org/A375786.

Commentary.

Write S = 3n². Ordered positive edges satisfy 3x² ≤ S and xS ≤ 2xyz + x³; hence x ≥ 3 forces xyz ≥ S. At x = 1 and n ≥ 2, the volume is strictly below S − 4. If no solution has first edge one, the explicit odd-n solution forces n to be even. Writing n = 2m then gives the solution (2, 2, 3m² − 1), of volume S − 4. With the first edge fixed, a smaller second edge gives no larger volume. These comparisons cover every competitor; n = 1 has only the unit cube.

References

• Truth anchor: D5/S3/Arith/OeisA375786CuboidMinimumVolume.claim
• Truth anchor: D5/S3/Arith/OeisA375786CuboidMinimumVolume.cuboids
• Truth anchor: D5/S3/Arith/OeisA375786CuboidMinimumVolume.result
• Truth anchor: D5/S3/Arith/OeisA375786CuboidMinimumVolume.volume