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erdos_699_ap_cofactors.json

json
{
  "source": "BFT Theorem 2.4 and (7.6), matching three or four prime-power exponents",
  "scope": "AP triples or two pairs with equal position sums; effective asymptotic bounds",
  "row_enumeration": false,
  "choices": [
    {
      "middle_prime": 2,
      "middle_power": 4,
      "endpoint_powers": {
        "3": 1,
        "5": 1
      },
      "seed": [
        16,
        15,
        1,
        1,
        11,
        8
      ],
      "gcd_certificate": {
        "c": 11,
        "d": 8,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1895,
        "positive_terms": 203,
        "largest_positive_t": 683,
        "ell_lower": "5899041743/15625000000"
      },
      "log_omega3": "[0.01909589458656980836150507034, 0.01909589458656980836150507034]",
      "log_omega4": "[2.037370591122885930638170755, 2.037370591122885930638170755]",
      "K": "[4.254312788510971005405779944, 4.254312788510971005405779944]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".348",
        "3": ".483",
        "5": ".243"
      },
      "alpha": {
        "2": "[0.348287523727139956961647202, 0.348287523727139956961647202]",
        "3": "[0.48353008192769529162056428, 0.48353008192769529162056428]",
        "5": "[0.243385245776716371628667294, 0.243385245776716371628667294]"
      }
    },
    {
      "middle_prime": 3,
      "middle_power": 4,
      "endpoint_powers": {
        "2": 4,
        "5": 1
      },
      "seed": [
        81,
        80,
        1,
        1,
        5,
        4
      ],
      "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"
      },
      "log_omega3": "[0.02244668197725239182290375167, 0.02244668197725239182290375167]",
      "log_omega4": "[3.907272323000387249904530942, 3.907272323000387249904530942]",
      "K": "[7.520267013072748565504605701, 7.520267013072748565504605701]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".262",
        "3": ".415",
        "5": ".571"
      },
      "alpha": {
        "2": "[0.2626355640244977202691078826, 0.2626355640244977202691078826]",
        "3": "[0.4156525098067115234870713542, 0.4156525098067115234870713542]",
        "5": "[0.5719731999844268809471248251, 0.5719731999844268809471248251]"
      }
    },
    {
      "middle_prime": 3,
      "middle_power": 4,
      "endpoint_powers": {
        "2": 3,
        "5": 1
      },
      "seed": [
        81,
        40,
        1,
        2,
        29,
        20
      ],
      "gcd_certificate": {
        "c": 29,
        "d": 20,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1959,
        "positive_terms": 264,
        "largest_positive_t": 881,
        "ell_lower": "423954159463/1000000000000"
      },
      "log_omega3": "[0.007314072553012716686794146086, 0.007314072553012716686794146086]",
      "log_omega4": "[2.629661919564519737904219462, 2.629661919564519737904219462]",
      "K": "[6.208009099199693757239063335, 6.208009099199693757239063335]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".3299",
        "3": ".292",
        "5": ".4813"
      },
      "alpha": {
        "2": "[0.3300778048318558728497081175, 0.3300778048318558728497081175]",
        "3": "[0.2921322948384612211553268275, 0.2921322948384612211553268275]",
        "5": "[0.4814962778834904551112621276, 0.4814962778834904551112621276]"
      }
    },
    {
      "middle_prime": 5,
      "middle_power": 2,
      "endpoint_powers": {
        "2": 3,
        "3": 1
      },
      "seed": [
        25,
        24,
        1,
        1,
        4,
        3
      ],
      "gcd_certificate": {
        "c": 4,
        "d": 3,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1714,
        "positive_terms": 160,
        "largest_positive_t": 503,
        "ell_lower": "376473179729/1000000000000"
      },
      "log_omega3": "[0.05641475950965783555925972818, 0.05641475950965783555925972818]",
      "log_omega4": "[2.590704269952757994847996479, 2.590704269952757994847996479]",
      "K": "[5.161904027332769245337516026, 5.161904027332769245337516026]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".194",
        "3": ".574",
        "5": ".376"
      },
      "alpha": {
        "2": "[0.1943122031448098954811805144, 0.1943122031448098954811805144]",
        "3": "[0.5743383515652930063142086451, 0.5743383515652930063142086451]",
        "5": "[0.3764169562580107438463567436, 0.3764169562580107438463567436]"
      }
    },
    {
      "middle_prime": 3,
      "middle_power": 3,
      "endpoint_powers": {
        "2": 2,
        "7": 1
      },
      "seed": [
        28,
        27,
        1,
        1,
        4,
        3
      ],
      "gcd_certificate": {
        "c": 4,
        "d": 3,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1714,
        "positive_terms": 160,
        "largest_positive_t": 503,
        "ell_lower": "376473179729/1000000000000"
      },
      "log_omega3": "[0.09491666902169909003906431627, 0.09491666902169909003906431627]",
      "log_omega4": "[2.709246407982561346419917371, 2.709246407982561346419917371]",
      "K": "[5.364139316162124933754755014, 5.364139316162124933754755014]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".483",
        "3": ".385",
        "7": ".274"
      },
      "alpha": {
        "2": "[0.4831251466780541506139716189, 0.4831251466780541506139716189]",
        "3": "[0.3855795549392259296360358322, 0.3855795549392259296360358322]",
        "7": "[0.274474418219416577261926523, 0.274474418219416577261926523]"
      }
    },
    {
      "middle_prime": 3,
      "middle_power": 5,
      "endpoint_powers": {
        "5": 1,
        "7": 2
      },
      "seed": [
        245,
        243,
        1,
        1,
        5,
        4
      ],
      "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"
      },
      "log_omega3": "[0.2996741018500753947903835286, 0.2996741018500753947903835286]",
      "log_omega4": "[3.632544052841007183985759879, 3.632544052841007183985759879]",
      "K": "[8.407293452817532732000072723, 8.407293452817532732000072723]",
      "exponent_margin": "1/10000",
      "targets": {
        "3": ".346",
        "5": ".617",
        "7": ".074"
      },
      "alpha": {
        "3": "[0.34663141305009865247605416, 0.34663141305009865247605416]",
        "5": "[0.6171329283398023602002503682, 0.6171329283398023602002503682]",
        "7": "[0.07417997957324482245495399904, 0.07417997957324482245495399904]"
      }
    },
    {
      "middle_prime": 2,
      "middle_power": 6,
      "endpoint_powers": {
        "3": 2,
        "7": 1
      },
      "seed": [
        64,
        63,
        1,
        1,
        4,
        3
      ],
      "gcd_certificate": {
        "c": 4,
        "d": 3,
        "terms": 4000,
        "m0": 100000000,
        "rounding_denominator": 1000000000000,
        "root_scale": 1000000,
        "eligible_terms": 1714,
        "positive_terms": 160,
        "largest_positive_t": 503,
        "ell_lower": "376473179729/1000000000000"
      },
      "log_omega3": "[0.373860509448379015681343878, 0.373860509448379015681343878]",
      "log_omega4": "[3.560033048072152905724447818, 3.560033048072152905724447818]",
      "K": "[6.828907869413786535796728592, 6.828907869413786535796728592]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".390",
        "3": ".356",
        "7": ".429"
      },
      "alpha": {
        "2": "[0.3909885500158779315413370357, 0.3909885500158779315413370357]",
        "3": "[0.3564931261768990362256214718, 0.3564931261768990362256214718]",
        "7": "[0.4300962360992121760884922277, 0.4300962360992121760884922277]"
      }
    },
    {
      "middle_prime": 7,
      "middle_power": 1,
      "endpoint_powers": {
        "2": 1,
        "3": 1
      },
      "seed": [
        7,
        6,
        1,
        1,
        3,
        2
      ],
      "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"
      },
      "log_omega3": "[0.03614975415760539448746210016, 0.03614975415760539448746210016]",
      "log_omega4": "[0.9865366657591167890638198473, 0.9865366657591167890638198473]",
      "K": "[2.603601259561391164481232642, 2.603601259561391164481232642]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".428",
        "3": ".13",
        "7": ".198"
      },
      "alpha": {
        "2": "[0.4675473611679581647147148324, 0.4675473611679581647147148324]",
        "3": "[0.1560825340411883808165641191, 0.1560825340411883808165641191]",
        "7": "[0.2526082318061537816934501942, 0.2526082318061537816934501942]"
      }
    }
  ],
  "four_factor_choices": [
    {
      "equal_position_sums": [
        [
          2,
          3
        ],
        [
          5,
          7
        ]
      ],
      "prime_powers": {
        "2": 2,
        "3": 2,
        "5": 1,
        "7": 1
      },
      "seed": [
        36,
        35,
        1,
        1,
        13,
        10
      ],
      "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"
      },
      "log_omega3": "[0.0186898452408733619097740844, 0.0186898452408733619097740844]",
      "log_omega4": "[2.994619678964844408307196583, 2.994619678964844408307196583]",
      "K": "[5.887072537659836469553567473, 5.887072537659836469553567473]",
      "exponent_margin": "1/10000",
      "targets": {
        "2": ".528",
        "3": ".253",
        "5": ".453",
        "7": ".338"
      },
      "alpha": {
        "2": "[0.529037784993573420883099984, 0.529037784993573420883099984]",
        "3": "[0.2535425499582393005134009781, 0.2535425499582393005134009781]",
        "5": "[0.4532298006730977992669236336, 0.4532298006730977992669236336]",
        "7": "[0.3389209537992783617251051379, 0.3389209537992783617251051379]"
      }
    }
  ]
}