Wiki
Wiki

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

erdos_699_fourteen_product_library.json

json
{
  "scope": "Conditional cofactor sources; no full-index effectivity assertion.",
  "height_variable": "h=n-13",
  "hypothesis": "The two unordered position pairs are different and have equal sums.",
  "records": [
    {
      "position_partition": [
        [
          2,
          5
        ],
        [
          3,
          13
        ]
      ],
      "seed": [
        40,
        39,
        1,
        1,
        13,
        10
      ],
      "powers": {
        "2": 3,
        "3": 1,
        "5": 1,
        "13": 1
      },
      "targets": {
        "2": "0.3139",
        "3": "0.6365",
        "5": "0.4685",
        "13": "0.1542"
      },
      "gcd_certificate": {
        "c": 13,
        "d": 10,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1913,
        "positive_terms": 213,
        "largest_positive_t": 712,
        "ell_lower": "334119934159/1000000000000"
      },
      "bound": {
        "seed": [
          40,
          39,
          1,
          1,
          13,
          10
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            3,
            1,
            0,
            "3139/10000"
          ],
          [
            5,
            1,
            1,
            0,
            "937/2000"
          ],
          [
            3,
            1,
            1,
            1,
            "1273/2000"
          ],
          [
            13,
            1,
            1,
            1,
            "771/5000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.315523824607675859627479, 0.315523824607675859627479]"
          },
          {
            "prime": 5,
            "exponent": "[0.470231846025170906819071, 0.470231846025170906819071]"
          },
          {
            "prime": 3,
            "exponent": "[0.638376976455376408240943, 0.638376976455376408240943]"
          },
          {
            "prime": 13,
            "exponent": "[0.155712391485645098769688, 0.155712391485645098769688]"
          }
        ],
        "log_omega3": "[0.0507193797095640150904057, 0.0507193797095640150904057]",
        "log_omega4": "[3.10326780452845632719197, 3.10326780452845632719197]",
        "K": "[6.0760085345204411699232, 6.0760085345204411699232]",
        "L": "[3.03257195254378899982678, 3.03257195254378899982678]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[3942343577.73036108284407, 3942343577.73036108284407]",
          "[22202.0053442538179868261, 22202.0053442538179868261]"
        ],
        "log_threshold": "[3942343577.73036108284407, 3942343577.73036108284407]",
        "threshold_power_of_ten": 1712138062
      }
    },
    {
      "position_partition": [
        [
          2,
          11
        ],
        [
          3,
          13
        ]
      ],
      "seed": [
        352,
        351,
        1,
        1,
        6,
        5
      ],
      "powers": {
        "2": 5,
        "3": 3,
        "11": 1,
        "13": 1
      },
      "targets": {
        "2": "0.3325",
        "3": "0.3651",
        "11": "0.5376",
        "13": "0.5055"
      },
      "gcd_certificate": {
        "c": 6,
        "d": 5,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1818,
        "positive_terms": 179,
        "largest_positive_t": 571,
        "ell_lower": "134020083931/500000000000"
      },
      "bound": {
        "seed": [
          352,
          351,
          1,
          1,
          6,
          5
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            5,
            1,
            0,
            "133/400"
          ],
          [
            11,
            1,
            1,
            0,
            "336/625"
          ],
          [
            3,
            3,
            1,
            1,
            "3651/10000"
          ],
          [
            13,
            1,
            1,
            1,
            "1011/2000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.334296430779105341085504, 0.334296430779105341085504]"
          },
          {
            "prime": 11,
            "exponent": "[0.539408804799506866519454, 0.539408804799506866519454]"
          },
          {
            "prime": 3,
            "exponent": "[0.366930883713190903126461, 0.366930883713190903126461]"
          },
          {
            "prime": 13,
            "exponent": "[0.507320814389342706778698, 0.507320814389342706778698]"
          }
        ],
        "log_omega3": "[0.0541877692266032925566131, 0.0541877692266032925566131]",
        "log_omega4": "[5.4583422593682212319957, 5.4583422593682212319957]",
        "K": "[10.4122497250716147844779, 10.4122497250716147844779]",
        "L": "[5.19211371205951542634631, 5.19211371205951542634631]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[3115268258.38839152816488, 3115268258.38839152816488]",
          "[11786.224458252099082712, 11786.224458252099082712]"
        ],
        "log_threshold": "[3115268258.38839152816488, 3115268258.38839152816488]",
        "threshold_power_of_ten": 1352943815
      }
    },
    {
      "position_partition": [
        [
          3,
          13
        ],
        [
          7,
          11
        ]
      ],
      "seed": [
        39,
        77,
        2,
        1,
        16,
        11
      ],
      "powers": {
        "3": 1,
        "7": 1,
        "11": 1,
        "13": 1
      },
      "targets": {
        "3": "0.6399",
        "7": "0.3638",
        "11": "0.2165",
        "13": "0.1621"
      },
      "gcd_certificate": {
        "c": 16,
        "d": 11,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1926,
        "positive_terms": 223,
        "largest_positive_t": 755,
        "ell_lower": "42814616731/100000000000"
      },
      "bound": {
        "seed": [
          39,
          77,
          2,
          1,
          16,
          11
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            3,
            1,
            1,
            0,
            "6399/10000"
          ],
          [
            13,
            1,
            1,
            0,
            "1621/10000"
          ],
          [
            7,
            1,
            1,
            1,
            "1819/5000"
          ],
          [
            11,
            1,
            1,
            1,
            "433/2000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 3,
            "exponent": "[0.641776072213177871760509, 0.641776072213177871760509]"
          },
          {
            "prime": 13,
            "exponent": "[0.163648319901749824211043, 0.163648319901749824211043]"
          },
          {
            "prime": 7,
            "exponent": "[0.365498107107538584747165, 0.365498107107538584747165]"
          },
          {
            "prime": 11,
            "exponent": "[0.218119556914237121844737, 0.218119556914237121844737]"
          }
        ],
        "log_omega3": "[0.0110389372988055816638084, 0.0110389372988055816638084]",
        "log_omega4": "[2.60342809102419321177762, 2.60342809102419321177762]",
        "K": "[6.13366223443281668226441, 6.13366223443281668226441]",
        "L": "[3.06116404996006293836196, 3.06116404996006293836196]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[4897862528.91472550074015, 4897862528.91472550074015]",
          "[26577.1291669203831731701, 26577.1291669203831731701]"
        ],
        "log_threshold": "[4897862528.91472550074015, 4897862528.91472550074015]",
        "threshold_power_of_ten": 2127114670
      }
    },
    {
      "position_partition": [
        [
          2,
          11
        ],
        [
          5,
          13
        ]
      ],
      "seed": [
        4225,
        704,
        1,
        6,
        10,
        7
      ],
      "powers": {
        "2": 6,
        "5": 2,
        "11": 1,
        "13": 2
      },
      "targets": {
        "2": "0.2989",
        "5": "0.4569",
        "11": "0.5949",
        "13": "0.1356"
      },
      "gcd_certificate": {
        "c": 10,
        "d": 7,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1882,
        "positive_terms": 197,
        "largest_positive_t": 662,
        "ell_lower": "81548571653/200000000000"
      },
      "bound": {
        "seed": [
          4225,
          704,
          1,
          6,
          10,
          7
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            6,
            1,
            1,
            "2989/10000"
          ],
          [
            11,
            1,
            1,
            1,
            "5949/10000"
          ],
          [
            5,
            2,
            1,
            0,
            "4569/10000"
          ],
          [
            13,
            2,
            1,
            0,
            "339/2500"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.300685877204800374155763, 0.300685877204800374155763]"
          },
          {
            "prime": 11,
            "exponent": "[0.596795102040446817342305, 0.596795102040446817342305]"
          },
          {
            "prime": 5,
            "exponent": "[0.458747630376771719503852, 0.458747630376771719503852]"
          },
          {
            "prime": 13,
            "exponent": "[0.13741008151721546146869, 0.13741008151721546146869]"
          }
        ],
        "log_omega3": "[0.0397876230023747004895594, 0.0397876230023747004895594]",
        "log_omega4": "[5.06486108968804186793147, 5.06486108968804186793147]",
        "K": "[11.8941773025729035288605, 11.8941773025729035288605]",
        "L": "[5.93463525557967777892986, 5.93463525557967777892986]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[5934635314.92603033472664, 5934635314.92603033472664]",
          "[28467.4578000988964054711, 28467.4578000988964054711]"
        ],
        "log_threshold": "[5934635314.92603033472664, 5934635314.92603033472664]",
        "threshold_power_of_ten": 2577379370
      }
    },
    {
      "position_partition": [
        [
          2,
          13
        ],
        [
          5,
          7
        ]
      ],
      "seed": [
        35,
        104,
        3,
        1,
        11,
        7
      ],
      "powers": {
        "2": 3,
        "5": 1,
        "7": 1,
        "13": 1
      },
      "targets": {
        "2": "0.2982",
        "5": "0.4564",
        "7": "0.3431",
        "13": "0.1348"
      },
      "gcd_certificate": {
        "c": 11,
        "d": 7,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1778,
        "positive_terms": 194,
        "largest_positive_t": 665,
        "ell_lower": "233950123267/500000000000"
      },
      "bound": {
        "seed": [
          35,
          104,
          3,
          1,
          11,
          7
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            3,
            1,
            1,
            "1491/5000"
          ],
          [
            13,
            1,
            1,
            1,
            "337/2500"
          ],
          [
            5,
            1,
            1,
            0,
            "1141/2500"
          ],
          [
            7,
            1,
            1,
            0,
            "3431/10000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.29986159098761755514465, 0.29986159098761755514465]"
          },
          {
            "prime": 13,
            "exponent": "[0.136393341031488935515657, 0.136393341031488935515657]"
          },
          {
            "prime": 5,
            "exponent": "[0.458109652601469981654179, 0.458109652601469981654179]"
          },
          {
            "prime": 7,
            "exponent": "[0.344820997112502712117313, 0.344820997112502712117313]"
          }
        ],
        "log_omega3": "[0.0118953414182397089229155, 0.0118953414182397089229155]",
        "log_omega4": "[2.03609392167223574229042, 2.03609392167223574229042]",
        "K": "[5.94008703111461359121727, 5.94008703111461359121727]",
        "L": "[2.96457392217006095244277, 2.96457392217006095244277]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[3261031346.99738019155772, 3261031346.99738019155772]",
          "[17805.1600582730910620606, 17805.1600582730910620606]"
        ],
        "log_threshold": "[3261031346.99738019155772, 3261031346.99738019155772]",
        "threshold_power_of_ten": 1416247920
      }
    },
    {
      "position_partition": [
        [
          2,
          13
        ],
        [
          3,
          5
        ]
      ],
      "seed": [
        676,
        675,
        1,
        1,
        13,
        11
      ],
      "powers": {
        "2": 2,
        "3": 3,
        "5": 2,
        "13": 2
      },
      "targets": {
        "2": "0.7611",
        "3": "0.4349",
        "5": "0.4481",
        "13": "0.1216"
      },
      "gcd_certificate": {
        "c": 13,
        "d": 11,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1833,
        "positive_terms": 210,
        "largest_positive_t": 719,
        "ell_lower": "121205621943/500000000000"
      },
      "bound": {
        "seed": [
          676,
          675,
          1,
          1,
          13,
          11
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            2,
            1,
            0,
            "7611/10000"
          ],
          [
            13,
            2,
            1,
            0,
            "76/625"
          ],
          [
            3,
            3,
            1,
            1,
            "4349/10000"
          ],
          [
            5,
            2,
            1,
            1,
            "4481/10000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.763109662481050111208754, 0.763109662481050111208754]"
          },
          {
            "prime": 13,
            "exponent": "[0.123401586201028884367513, 0.123401586201028884367513]"
          },
          {
            "prime": 3,
            "exponent": "[0.436806547373929683487378, 0.436806547373929683487378]"
          },
          {
            "prime": 5,
            "exponent": "[0.449957669907400429748823, 0.449957669907400429748823]"
          }
        ],
        "log_omega3": "[0.0407447340941293200782911, 0.0407447340941293200782911]",
        "log_omega4": "[6.1311653369211648090767, 6.1311653369211648090767]",
        "K": "[11.7041022072839496986217, 11.7041022072839496986217]",
        "L": "[5.84004862810003810576841, 5.84004862810003810576841]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[7592063292.45068170279943, 7592063292.45068170279943]",
          "[37206.8448992097306985629, 37206.8448992097306985629]"
        ],
        "log_threshold": "[7592063292.45068170279943, 7592063292.45068170279943]",
        "threshold_power_of_ten": 3297191195
      }
    },
    {
      "position_partition": [
        [
          2,
          11
        ],
        [
          7,
          13
        ]
      ],
      "seed": [
        91,
        88,
        1,
        1,
        5,
        4
      ],
      "powers": {
        "2": 3,
        "7": 1,
        "11": 1,
        "13": 1
      },
      "targets": {
        "2": "0.2998",
        "7": "0.3447",
        "11": "0.1929",
        "13": "0.1369"
      },
      "gcd_certificate": {
        "c": 5,
        "d": 4,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1778,
        "positive_terms": 171,
        "largest_positive_t": 539,
        "ell_lower": "311676144201/1000000000000"
      },
      "bound": {
        "seed": [
          91,
          88,
          1,
          1,
          5,
          4
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            3,
            1,
            1,
            "1499/5000"
          ],
          [
            11,
            1,
            1,
            1,
            "1929/10000"
          ],
          [
            7,
            1,
            1,
            0,
            "3447/10000"
          ],
          [
            13,
            1,
            1,
            0,
            "1369/10000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.301381303704132775779158, 0.301381303704132775779158]"
          },
          {
            "prime": 11,
            "exponent": "[0.194392130887636554507198, 0.194392130887636554507198]"
          },
          {
            "prime": 7,
            "exponent": "[0.346243121437443557794249, 0.346243121437443557794249]"
          },
          {
            "prime": 13,
            "exponent": "[0.138267876130274607694933, 0.138267876130274607694933]"
          }
        ],
        "log_omega3": "[0.0489458281455525415702794, 0.0489458281455525415702794]",
        "log_omega4": "[1.8026863242512737924216, 1.8026863242512737924216]",
        "K": "[5.95300856591786907509612, 5.95300856591786907509612]",
        "L": "[2.97178699740648445840979, 2.97178699740648445840979]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[1485893513.56217721623732, 1485893513.56217721623732]",
          "[9386.42127222247791856415, 9386.42127222247791856415]"
        ],
        "log_threshold": "[1485893513.56217721623732, 1485893513.56217721623732]",
        "threshold_power_of_ten": 645315354
      }
    },
    {
      "position_partition": [
        [
          3,
          11
        ],
        [
          5,
          13
        ]
      ],
      "seed": [
        33,
        65,
        2,
        1,
        3,
        2
      ],
      "powers": {
        "3": 1,
        "5": 1,
        "11": 1,
        "13": 1
      },
      "targets": {
        "3": "0.6172",
        "5": "0.4402",
        "11": "0.167",
        "13": "0.1091"
      },
      "gcd_certificate": {
        "c": 3,
        "d": 2,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1600,
        "positive_terms": 148,
        "largest_positive_t": 454,
        "ell_lower": "24308690901/50000000000"
      },
      "bound": {
        "seed": [
          33,
          65,
          2,
          1,
          3,
          2
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            3,
            1,
            1,
            0,
            "1543/2500"
          ],
          [
            11,
            1,
            1,
            0,
            "167/1000"
          ],
          [
            5,
            1,
            1,
            1,
            "2201/5000"
          ],
          [
            13,
            1,
            1,
            1,
            "1091/10000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 3,
            "exponent": "[0.618999596163538536294877, 0.618999596163538536294877]"
          },
          {
            "prime": 11,
            "exponent": "[0.168406291539563328822225, 0.168406291539563328822225]"
          },
          {
            "prime": 5,
            "exponent": "[0.441844496996747004544315, 0.441844496996747004544315]"
          },
          {
            "prime": 13,
            "exponent": "[0.11047168223680440261071, 0.11047168223680440261071]"
          }
        ],
        "log_omega3": "[0.151187399300013852805717, 0.151187399300013852805717]",
        "log_omega4": "[2.38890038813442821858678, 2.38890038813442821858678]",
        "K": "[5.7669875286519225897121, 5.7669875286519225897121]",
        "L": "[2.87905416709118502194802, 2.87905416709118502194802]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[863716258.76451800785796, 863716258.76451800785796]",
          "[5969.96768238281534327349, 5969.96768238281534327349]"
        ],
        "log_threshold": "[863716258.76451800785796, 863716258.76451800785796]",
        "threshold_power_of_ten": 375107206
      }
    },
    {
      "position_partition": [
        [
          2,
          11
        ],
        [
          5,
          7
        ]
      ],
      "seed": [
        176,
        175,
        1,
        1,
        11,
        9
      ],
      "powers": {
        "2": 4,
        "5": 2,
        "7": 1,
        "11": 1
      },
      "targets": {
        "2": "0.3841",
        "5": "0.2853",
        "7": "0.5671",
        "11": "0.4671"
      },
      "gcd_certificate": {
        "c": 11,
        "d": 9,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1800,
        "positive_terms": 199,
        "largest_positive_t": 679,
        "ell_lower": "275101334003/1000000000000"
      },
      "bound": {
        "seed": [
          176,
          175,
          1,
          1,
          11,
          9
        ],
        "m0": 100000000,
        "gap": 169,
        "degree": 2,
        "maximum_shift": 13,
        "components": [
          [
            2,
            4,
            1,
            0,
            "3841/10000"
          ],
          [
            11,
            1,
            1,
            0,
            "4671/10000"
          ],
          [
            5,
            2,
            1,
            1,
            "2853/10000"
          ],
          [
            7,
            1,
            1,
            1,
            "5671/10000"
          ]
        ],
        "limiting_exponents": [
          {
            "prime": 2,
            "exponent": "[0.385897220647692097063873, 0.385897220647692097063873]"
          },
          {
            "prime": 11,
            "exponent": "[0.468888357003895615751464, 0.468888357003895615751464]"
          },
          {
            "prime": 5,
            "exponent": "[0.287048751736730736157314, 0.287048751736730736157314]"
          },
          {
            "prime": 7,
            "exponent": "[0.568998884934010898537635, 0.568998884934010898537635]"
          }
        ],
        "log_omega3": "[0.0371713048475053770637688, 0.0371713048475053770637688]",
        "log_omega4": "[4.7168465848530139826845, 4.7168465848530139826845]",
        "K": "[9.02972210991789049483637, 9.02972210991789049483637]",
        "L": "[4.50381394272869840380792, 4.50381394272869840380792]",
        "M": "[100000000.0, 100000000.0]",
        "log_threshold_terms": [
          "[2.56494935746153673605349, 2.56494935746153673605349]",
          "[4954195386.54352161420439, 4954195386.54352161420439]",
          "[20405.3831297170188631249, 20405.3831297170188631249]"
        ],
        "log_threshold": "[4954195386.54352161420439, 4954195386.54352161420439]",
        "threshold_power_of_ten": 2151579719
      }
    }
  ],
  "uniform_threshold_power_of_ten": 3297191195
}