bibkey: oeis2026a038867 authors: OEIS Foundation Inc. year: 2026 title: OEIS A038867 doi: null url: https://oeis.org/A038867 claim: A comment on OEIS A038867 conjectures that no term of the sequence of cube differences at gap five is a perfect square. strata_touched:

• D5/S3/Factorization/CubeGapFiveNoSquare license: citation-only triage: anchor

OEIS A038867

The sequence lists the difference between two cubes whose bases are five apart. Klaus Purath added a comment on May 15, 2026 conjecturing that no term is a perfect square, noting verification through one hundred thousand terms. The caller found the statement still labelled Conjecture on September 9, 2026.

The polynomial form already on the page

A formula added by Oliveira on September 12, 2025 expands the difference as five times a quadratic. That expansion is the whole of what the page supplies; it names the terms but says nothing about squares. What the conjecture needs beyond it is one further step, and that step is short: the expansion is two modulo three for every index, whereas a square is zero or one modulo three. The two residue sets do not meet.

This is a different situation from an entry whose printed description already determines the conjectured property. Here the printed formula is a polynomial identity and the residue exclusion is not stated anywhere on the page.

What the Lean module proves

D5/S3/Factorization/CubeGapFiveNoSquare states the conjecture for every index and every candidate square. It also records the polynomial identity separately, because subtraction of naturals truncates and the identity is what shows the subtraction here does not.

All four public results depend only on propositional extensionality and quotient soundness; no choice principle enters.

ASSUMED-UNVERIFIED: the identification of the printed sequence with the module’s definition is a human reading of the entry, not a machine proof that the two denote the same terms.

Search log

• Caller reading, 2026-09-09: the seat reported checking the entry’s comments, links, formula and cross-reference sections, its sole cross-reference, and searches pairing the A-number with the words proof and perfect square, with the author’s name, and with both the cube-difference and the expanded polynomial written out, restricted in turn to arXiv, MathOverflow and Mathematics Stack Exchange. Relevant hits returned to the entry itself.
• The seat’s receipt is 所查来源未找到, explicitly not a claim that no proof exists anywhere.
• Caller verification, 2026-09-09: the orchestrator checked the polynomial identity, both residue sets, and their disjointness, and searched three hundred thousand terms directly. Readings are in the problem entry. The seat’s written derivation contained a compressed step that read as five times twenty-five; the orchestrator recomputed and confirmed the conclusion while noting the intermediate reduction is of the quadratic, not of the constant alone.

Verified locator

• URL: https://oeis.org/A038867
• Cross-referenced entry: https://oeis.org/A378569