files / problems / two-smooth-summands / compute / certs / trivial_cover.json

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{
  "range": [
    2,
    200000
  ],
  "failures_in_range": [],
  "worst_ratio_in_range": {
    "n": 197136,
    "F_split_over_sqrt": 1.9977477477477477
  },
  "large_spot": [
    {
      "n": 1000000000000,
      "a": 999998000001,
      "b": 1999999,
      "P_a": 37,
      "P_b": 1657,
      "F_split": 1657,
      "bound": 2000001.0,
      "ok": true
    },
    {
      "n": 1000000000007,
      "a": 1000000000000,
      "b": 7,
      "P_a": 5,
      "P_b": 7,
      "F_split": 7,
      "bound": 2000001.000007,
      "ok": true
    },
    {
      "n": 1000000000000000003,
      "a": 1000000000000000000,
      "b": 3,
      "P_a": 5,
      "P_b": 3,
      "F_split": 5,
      "bound": 2000000001.0,
      "ok": true
    },
    {
      "n": 9998200081,
      "a": 9998000100,
      "b": 199981,
      "P_a": 101,
      "P_b": 6451,
      "F_split": 6451,
      "bound": 199983.0,
      "ok": true
    },
    {
      "n": 131486759,
      "a": 131469156,
      "b": 17603,
      "P_a": 13,
      "P_b": 607,
      "F_split": 607,
      "bound": 22934.535183220225,
      "ok": true
    }
  ],
  "extra_failures": [],
  "statement": "F(n) < 2*sqrt(n)+1 for every n>=2 via n = m^2+r or (m-1)^2+(2m-1).",
  "is_dent": false,
  "reason": "Balog 1989 already calls the N^{1/2} statement almost trivial."
}