{
  "state": "passed_two_complete_equation_certificates",
  "domain": "m,n positive integers",
  "absolute_difference_limit": 20,
  "arithmetic": "Exact integers and fractions; 100 atanh-series terms with a geometric remainder.",
  "logarithm_intervals": {
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      "programme_input": "Lesson 7 Theorem 7.1; Lesson 11 Theorem 11.3 and (11.13)",
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          1
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        [
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      "gcd_handling": "For m=g*p,n=g*q the logarithmic error is multiplied by g>=1.",
      "bound_proof": "The positive endpoint and derivative margins exclude m at or above its bound; Legendre plus every relevant convergent error excludes n above the finite search bound."
    },
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          "error_lower_exact": "706899220142904066306698837819542844936894179528666579123500474643184995763079980680230858886258805678030736305183595445669801586241648806376351691215236047776217529249839/21088470356099387607271664275583996808574600365019163073413640018743604357750275714193373080770383124228088463689660795073416411849275013791586034388669960401102764949856411746500",
          "error_upper_exact": "176724805035726016576679834122455063369093320289366844645816208098394383902428303151513545297445727475362042627254637418327482813415719634909161394702566717553172917055716/5272117589024846901817916068895999202143650091254790768353410004685901089437568928548343270192595781057022115922415198768354102962318753447896508597167490100275691237464102936625",
          "L_scaled_by_100000000": 3
        }
      ],
      "uniform_error_strict_lower_bound": "1/100000000",
      "n_upper_bound": 13,
      "small_n_exhaustive_rows": [
        {
          "n": 1,
          "all_m_checked": [
            1,
            4
          ],
          "rows": [
            {
              "m": 1,
              "difference": -3
            },
            {
              "m": 2,
              "difference": -1
            },
            {
              "m": 3,
              "difference": 3
            },
            {
              "m": 4,
              "difference": 11
            }
          ]
        },
        {
          "n": 2,
          "all_m_checked": [
            1,
            5
          ],
          "rows": [
            {
              "m": 1,
              "difference": -23
            },
            {
              "m": 2,
              "difference": -21
            },
            {
              "m": 3,
              "difference": -17
            },
            {
              "m": 4,
              "difference": -9
            },
            {
              "m": 5,
              "difference": 7
            }
          ]
        },
        {
          "n": 3,
          "all_m_checked": [
            1,
            7
          ],
          "rows": [
            {
              "m": 1,
              "difference": -123
            },
            {
              "m": 2,
              "difference": -121
            },
            {
              "m": 3,
              "difference": -117
            },
            {
              "m": 4,
              "difference": -109
            },
            {
              "m": 5,
              "difference": -93
            },
            {
              "m": 6,
              "difference": -61
            },
            {
              "m": 7,
              "difference": 3
            }
          ]
        }
      ],
      "nearest_power_rows": [
        {
          "n": 4,
          "k": 9,
          "distance_below": 113,
          "distance_above": 399
        },
        {
          "n": 5,
          "k": 11,
          "distance_below": 1077,
          "distance_above": 971
        },
        {
          "n": 6,
          "k": 13,
          "distance_below": 7433,
          "distance_above": 759
        },
        {
          "n": 7,
          "k": 16,
          "distance_below": 12589,
          "distance_above": 52947
        },
        {
          "n": 8,
          "k": 18,
          "distance_below": 128481,
          "distance_above": 133663
        },
        {
          "n": 9,
          "k": 20,
          "distance_below": 904549,
          "distance_above": 144027
        },
        {
          "n": 10,
          "k": 23,
          "distance_below": 1377017,
          "distance_above": 7011591
        },
        {
          "n": 11,
          "k": 25,
          "distance_below": 15273693,
          "distance_above": 18280739
        },
        {
          "n": 12,
          "k": 27,
          "distance_below": 109922897,
          "distance_above": 24294831
        },
        {
          "n": 13,
          "k": 30,
          "distance_below": 146961301,
          "distance_above": 926780523
        }
      ],
      "solutions_m_n_signed_difference": [
        [
          1,
          1,
          -3
        ],
        [
          2,
          1,
          -1
        ],
        [
          3,
          1,
          3
        ],
        [
          4,
          1,
          11
        ],
        [
          3,
          2,
          -17
        ],
        [
          4,
          2,
          -9
        ],
        [
          5,
          2,
          7
        ],
        [
          7,
          3,
          3
        ]
      ],
      "gcd_handling": "For m=g*p,n=g*q the logarithmic error is multiplied by g>=1.",
      "bound_proof": "The positive endpoint and derivative margins exclude m at or above its bound; Legendre plus every relevant convergent error excludes n above the finite search bound."
    }
  ]
}