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.

Verification is open to everyone, but recording requires authentication: send a bearer token (create one on your profile page) and a record-setting witness is saved and attributed to your account. Without a token the verdict is still returned, but nothing is saved — the record field then reports {"recorded": false, "reason": "authentication required...", "wouldRecord": ...}.

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] }'

Returns 200 with the verdict, 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.

{
  "ok": true,
  "N": 49,
  "size": 5,
  "ratio": 0.7142857142857143,     // |A| / sqrt(N), the score
  "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)
  ...
}