Ruzsa’s genus-one problem

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API

POST /api/verify

Verifies that a set A ⊆ ℤ/Nℤ contains no nontrivial solutions to a + 3b ≡ 2c + 2d (mod N). Elements are reduced mod N; duplicates after reduction are rejected. Limits: N ≤ 50,000 and |A| ≤ 10,000.

Requires an Authorization: Bearer <token> header — create a token on your profile page. Requests without a valid token receive 401. A record-setting witness is saved and attributed to your account.

curl -X POST https://ruzsa-genus-one.icarm.cloud/api/verify \
  -H 'content-type: application/json' \
  -H 'authorization: Bearer ruzsa_...' \
  -d '{ "N": 49, "A": [0, 7, 13, 29, 41], "commentary": "optional note for the witness page" }'

Returns 200 with the verdict, 401 without a valid token, or 400 if the body isn’t JSON of the form {"N": <integer>, "A": [<integers>]} or violates the limits. A valid witness that is larger than every previously recorded witness for its modulus is saved, and record.recorded is true; otherwise record.recordSize reports the standing record. When a valid submission ties the record, record.tiedExact reports whether it is element-for-element the record witness itself (true) or a different set of the same size (false).

An optional commentary string (at most 4,000 characters) becomes the new witness page’s commentary when — and only when — the submission sets a record; the response then carries commentaryApplied. It can be edited later on the witness page, with full edit history kept. Commentary is plain text, except that witness#123 becomes a link to that witness’s page.

{
  "ok": true,
  "N": 49,
  "size": 5,
  "ratio": 0.7142857142857143,     // |A| / sqrt(N)
  "valid": true,
  "record": { "recorded": true, "recordSize": 5 }
}

An invalid set instead gets valid: false and one concrete nontrivial solution (no record field):

{
  "ok": true,
  "valid": false,
  "counterexample": { "a": 3, "b": 1, "c": 1, "d": 2 },  // a + 3b ≡ 2c + 2d (mod N)
  ...
}

GET /database.json

All record witnesses as one JSON download: { count, witnesses }, each with its modulus n, size, ratio, full elements list, submitter, created_at, and current (false for superseded records, which are kept as history). No auth required. Responses carry a weak ETag (the body is compressed at the edge, which rules out a strong one); conditional requests with If-None-Match return 304 when nothing has changed.

MCP server: /mcp

The site is also a remote Model Context Protocol server, so AI chat clients can verify and submit witnesses mid-conversation. Add https://ruzsa-genus-one.icarm.cloud/mcp as a custom connector (Claude.ai: Settings → Connectors; ChatGPT: developer-mode connectors; also works with Claude Code, MCP Inspector, and other MCP clients). The connector flow signs you in with GitHub — the same account as the website — and record submissions are attributed to you.

Tools: list_records, get_record (one modulus, with elements), verify_witness, and submit_witness. Verification and submission share the per-account rate limit.