Partial result for Problem 411: r=2 first branch fully classified

[Tonycollatz]

Updated informal and formal status for problem 411, marking it as 'partial proved' with a new last update date. Added a note regarding the solution.

[Tonycollatz]

The proof is available at Zenodo: https://doi.org/10.5281/zenodo.21991040

[Tonycollatz]

This PR resolves the special case r=2 only. The general case r > 2 remains open.

[Tonycollatz]

To close the necessity part: for each m in {1,3,5,7,35,47}, we check the candidates with exponent a < a_m, where a_1=2 and a_m=1 for others. Direct computation shows:
n=1,2 fail for m=1; n=3 fail for m=3; n=5 fail for m=5; n=7 fail for m=7; n=35 fail for m=35; n=47 fail for m=47.
Thus for each m, n=2^a m is a solution iff a >= a_m. Combined with the base cases and the doubling lemma, Branch 1 is fully classified.

[teorth]

Thanks for this, and for the follow-up comments spelling out the Branch 1 argument. A few things need resolving before it can be merged.

It fails schema validation. The PR adds a notes: key, but schema/problems.schema.json is additionalProperties: false and the only free-text field is comments. Merging the branch locally and running the repo's own check gives:

[schema] at '410': Additional properties are not allowed ('notes' was unexpected)
❌ Validation failed with one or more errors.

You won't have seen this in the PR checks: because this is your first contribution here, GitHub holds workflow runs pending maintainer approval, so Validate data never actually ran.

The description and the diff disagree. The title says "Add proof for the r=2 case", and the body says the problem is being marked proved — but the diff leaves informal_status.state: "open", and the note itself says "This is a partial result, not a complete proof of the r=2 case." Those three should be made to say the same thing. Based on your own description, open is the correct status, so it's the title and body that need changing.

The last_update bumps aren't supported by the diff. informal_status, status and formalized all move to 2026-08-18, but no state changes. last_update should track when the state it sits under actually changed. formalized in particular is autogenerated from the formal-conjectures repository and shouldn't be hand-edited at all.

The prose belongs on the problem page, not here. Even with notes renamed to comments, CONTRIBUTING.md asks that comments stay a very brief label — it renders as the final column of the README table — and directs detailed mathematical commentary to the problem's own page on erdosproblems.com/411. That page is the right home for the solution families, the branch structure, and the Zenodo link, and it's where subject-matter readers will actually find them.

So the shape that would work here is: post the mathematical content to the erdosproblems.com page for 411, and reduce the database change to whatever the status genuinely warrants — which, for a partial classification of one branch, may well be no change at all.

Two smaller points: the diff removes the blank line before - number: "412" and adds a line of trailing whitespace, both of which should go. Note also that whether a partial result of this kind warrants any database change is Terry's call, not something validation can settle.

[Tonycollatz]

Partial result for Problem 411 (r=2): Steinerberger's first branch (odd part in S) is fully classified and verified; the second branch (8m+7/6m+5) remains open. Proof: https://doi.org/10.5281/zenodo.21991040. Status: open.

[Tonycollatz]

I have verified the first branch of Steinerberger's reduction for the r=2 case.

The solution set for this branch is:
{ 2^a · m | m ∈ {1,3,5,7,35,47}, a ≥ a_m }
where a_1=2 and a_m=1 for m∈{3,5,7,35,47}.

Verification:

1. Direct base-case check: n=4,6,10,14,70,94 all satisfy g_{k+2}=2g_k.
2. Doubling lemma: if n satisfies the condition, then 2n also does, using φ(2x)=2φ(x).

The second branch of Steinerberger's reduction (8m+7 and 6m+5 forms) remains open.
Full proof: https://doi.org/10.5281/zenodo.21991040

[Tonycollatz]

Thanks for taking the time to read through the argument and for the specific feedback. I’ve made the requested changes and left the detailed proof summary in the comments. Please let me know if anything still needs adjustment.

[teorth]

Thanks for reworking this — it is much closer. Moving from notes to comments fixes the schema failure, and dropping the last_update bumps and the proved framing was right: the status now matches what the result actually establishes.

Three things still stand in the way, all small.

The comment is too long for the field. comments renders as the final column of the README table and in practice holds short labels — across the 68 entries that use it the median is 21 characters and the longest is 34 ("second Hardy-Littlewood conjecture"). This one is 120. CONTRIBUTING.md asks that detailed mathematical commentary go to the problem's own page on erdosproblems.com/411 instead, and that is the better home for the branch structure and the Zenodo link — it is where readers interested in the mathematics will actually look, and it has room to state the result properly.

If you want a marker in the database pointing at that discussion, something in the field's existing register would work — for example "partial result on r=2" — with the substance on the problem page.

Field order. The new key goes after tags; all 68 existing entries put comments before tags. Worth matching so the file stays uniform.

Trailing whitespace. The diff replaces the blank line before - number: "412" with a line containing two spaces. That should be a plain empty line.

Separately, and not something validation can settle: whether a partial classification of one branch warrants a database annotation at all is Terry's call rather than mine.

[Tonycollatz]

Thanks again for the detailed review and for taking the time to look through this.
I've made the three changes you pointed out:
shortened comments to a brief label ("partial r=2 result")
moved comments before tags to match the existing entry order
removed the trailing whitespace (the blank line before - number: "411" is now clean)
The full mathematical details — branch structure, verification steps, and the Zenodo link — remain in the PR comments as a permanent record.
I appreciate your consideration of whether this partial result is worth including. Regardless of the final decision, I'm grateful for the feedback and the opportunity to contribute.
If there's anything else that needs adjustment, please let me know.