bibkey: geisser2021squares
authors: Michael Geißer, Theresa Körner, Sascha Kurz and Anne Zahn
year: 2021
title: "Squares with three digits"
doi: null
url: https://arxiv.org/abs/2112.00444v2
claim: "If ⟨a₁′,...,aₗ′⟩ is a solution with l ≥ 2, then ⟨a₁′,...,aₗ₋₁′⟩ is also a solution."
strata_touched:
  - D5/S1/Digit/SquaresThreeDigitsPrefixRefutation
license: citation-only
triage: anchor

Squares with three digits: prefix closure

Verified locator

• URL: https://arxiv.org/abs/2112.00444v2
• Versioned text: https://arxiv.org/html/2112.00444v2#Thmconjecture1
• PDF: https://arxiv.org/pdf/2112.00444v2
• Locator: Section 3.3, decimal block shorthand on printed page 25 and Conjecture 1 on printed page 26; governing equations (24)–(27).
• Original version: https://arxiv.org/abs/2112.00444v1, Section 3.3, Conjecture 1, with the same block and deletion conventions.

The bibliographic year is the first submission year. Version 2 is the revision of 10 January 2022; the versioned arXiv metadata identifies the four authors listed above. Only short quotations and citations are retained.

Conjecture 1 states:

If ⟨a₁′,…,aₗ′⟩ is a solution with l ≥ 2, then ⟨a₁′,…,aₗ₋₁′⟩ is also a solution.

Each entry is a nonnegative number of nines following a digit 4; concatenate these blocks and append a final 5. A zero entry gives a lone 4. A solution has every actual decimal digit of its ordinary integer square in {0,2,5}. The notation expresses subset membership, not occurrence of all three digits. The conjecture deletes the final whole block, retaining the final digit 5.

Source and literature boundary

Both named versions state the same conjecture. The versioned v2 text and arXiv metadata retain it as a conjecture. This source check establishes the published question and its literal domain; it is not an exhaustive search for subsequent resolutions. No worldwide openness, originality, priority or historical solved-credit conclusion follows from this note.

The repository result is the literal negation of that universal implication, with tuple [0,1,3,7,15,31,63,127,2,6,14,62,14,6,2]. Its definitions and proof are in D5/S1/Digit/SquaresThreeDigitsPrefixRefutation.lean; the formal exposition is its owning Blueprint. No general carry classification, least counterexample or assertion about other bases is attributed to it.