files / problems / two-smooth-summands / compute / q1 / certs / infinite_family.json

10 KB · on GitHub

{
  "floor_divisor_family": [
    {
      "exponent": "2/5",
      "n_primes_tested": 129,
      "n_hits": 129,
      "n_misses": 0,
      "sample_hits": [
        {
          "prime": 11,
          "u": 2,
          "n": 23,
          "u0": 3,
          "a": 21,
          "b": 2,
          "P_a": 7,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 13,
          "u": 4,
          "n": 53,
          "u0": 4,
          "a": 52,
          "b": 1,
          "P_a": 13,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 17,
          "u": 4,
          "n": 69,
          "u0": 5,
          "a": 65,
          "b": 4,
          "P_a": 13,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 19,
          "u": 3,
          "n": 58,
          "u0": 5,
          "a": 55,
          "b": 3,
          "P_a": 11,
          "P_b": 3,
          "P_exceeds": true
        },
        {
          "prime": 23,
          "u": 2,
          "n": 47,
          "u0": 4,
          "a": 44,
          "b": 3,
          "P_a": 11,
          "P_b": 3,
          "P_exceeds": true
        },
        {
          "prime": 29,
          "u": 2,
          "n": 59,
          "u0": 5,
          "a": 55,
          "b": 4,
          "P_a": 11,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 31,
          "u": 5,
          "n": 156,
          "u0": 7,
          "a": 154,
          "b": 2,
          "P_a": 11,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 37,
          "u": 2,
          "n": 75,
          "u0": 5,
          "a": 70,
          "b": 5,
          "P_a": 7,
          "P_b": 5,
          "P_exceeds": true
        },
        {
          "prime": 10267,
          "u": 432,
          "n": 4435345,
          "u0": 455,
          "a": 4435340,
          "b": 5,
          "P_a": 2437,
          "P_b": 5,
          "P_exceeds": true
        },
        {
          "prime": 10271,
          "u": 438,
          "n": 4498699,
          "u0": 458,
          "a": 4498476,
          "b": 223,
          "P_a": 1637,
          "P_b": 223,
          "P_exceeds": true
        },
        {
          "prime": 10273,
          "u": 432,
          "n": 4437937,
          "u0": 455,
          "a": 4437615,
          "b": 322,
          "P_a": 3251,
          "P_b": 23,
          "P_exceeds": true
        },
        {
          "prime": 10289,
          "u": 433,
          "n": 4455138,
          "u0": 456,
          "a": 4455120,
          "b": 18,
          "P_a": 977,
          "P_b": 3,
          "P_exceeds": true
        }
      ],
      "misses": []
    },
    {
      "exponent": "1/3",
      "n_primes_tested": 129,
      "n_hits": 129,
      "n_misses": 0,
      "sample_hits": [
        {
          "prime": 11,
          "u": 2,
          "n": 23,
          "u0": 2,
          "a": 22,
          "b": 1,
          "P_a": 11,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 13,
          "u": 3,
          "n": 40,
          "u0": 3,
          "a": 39,
          "b": 1,
          "P_a": 13,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 17,
          "u": 2,
          "n": 35,
          "u0": 3,
          "a": 33,
          "b": 2,
          "P_a": 11,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 19,
          "u": 3,
          "n": 58,
          "u0": 3,
          "a": 57,
          "b": 1,
          "P_a": 19,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 23,
          "u": 2,
          "n": 47,
          "u0": 3,
          "a": 45,
          "b": 2,
          "P_a": 5,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 29,
          "u": 2,
          "n": 59,
          "u0": 3,
          "a": 57,
          "b": 2,
          "P_a": 19,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 31,
          "u": 2,
          "n": 63,
          "u0": 3,
          "a": 60,
          "b": 3,
          "P_a": 5,
          "P_b": 3,
          "P_exceeds": true
        },
        {
          "prime": 37,
          "u": 4,
          "n": 149,
          "u0": 5,
          "a": 145,
          "b": 4,
          "P_a": 29,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 10267,
          "u": 61,
          "n": 626288,
          "u0": 85,
          "a": 626280,
          "b": 8,
          "P_a": 307,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 10271,
          "u": 62,
          "n": 636803,
          "u0": 86,
          "a": 636744,
          "b": 59,
          "P_a": 617,
          "P_b": 59,
          "P_exceeds": true
        },
        {
          "prime": 10273,
          "u": 61,
          "n": 626654,
          "u0": 85,
          "a": 626620,
          "b": 34,
          "P_a": 97,
          "P_b": 17,
          "P_exceeds": true
        },
        {
          "prime": 10289,
          "u": 61,
          "n": 627630,
          "u0": 85,
          "a": 627555,
          "b": 75,
          "P_a": 107,
          "P_b": 5,
          "P_exceeds": true
        }
      ],
      "misses": []
    },
    {
      "exponent": "27/100",
      "n_primes_tested": 129,
      "n_hits": 129,
      "n_misses": 0,
      "sample_hits": [
        {
          "prime": 11,
          "u": 2,
          "n": 23,
          "u0": 2,
          "a": 22,
          "b": 1,
          "P_a": 11,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 13,
          "u": 2,
          "n": 27,
          "u0": 2,
          "a": 26,
          "b": 1,
          "P_a": 13,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 17,
          "u": 2,
          "n": 35,
          "u0": 2,
          "a": 34,
          "b": 1,
          "P_a": 17,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 19,
          "u": 2,
          "n": 39,
          "u0": 2,
          "a": 38,
          "b": 1,
          "P_a": 19,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 23,
          "u": 2,
          "n": 47,
          "u0": 2,
          "a": 46,
          "b": 1,
          "P_a": 23,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 29,
          "u": 2,
          "n": 59,
          "u0": 3,
          "a": 57,
          "b": 2,
          "P_a": 19,
          "P_b": 2,
          "P_exceeds": true
        },
        {
          "prime": 31,
          "u": 2,
          "n": 63,
          "u0": 3,
          "a": 60,
          "b": 3,
          "P_a": 5,
          "P_b": 3,
          "P_exceeds": true
        },
        {
          "prime": 37,
          "u": 3,
          "n": 112,
          "u0": 3,
          "a": 111,
          "b": 1,
          "P_a": 37,
          "P_b": 1,
          "P_exceeds": true
        },
        {
          "prime": 10267,
          "u": 2,
          "n": 20535,
          "u0": 14,
          "a": 20524,
          "b": 11,
          "P_a": 733,
          "P_b": 11,
          "P_exceeds": true
        },
        {
          "prime": 10271,
          "u": 2,
          "n": 20543,
          "u0": 14,
          "a": 20538,
          "b": 5,
          "P_a": 163,
          "P_b": 5,
          "P_exceeds": true
        },
        {
          "prime": 10273,
          "u": 2,
          "n": 20547,
          "u0": 14,
          "a": 20538,
          "b": 9,
          "P_a": 163,
          "P_b": 3,
          "P_exceeds": true
        },
        {
          "prime": 10289,
          "u": 2,
          "n": 20579,
          "u0": 14,
          "a": 20566,
          "b": 13,
          "P_a": 113,
          "P_b": 13,
          "P_exceeds": true
        }
      ],
      "misses": []
    }
  ],
  "pow2_family": [
    {
      "k": 4,
      "pow2": 16,
      "q": 11,
      "n": 27,
      "a": 16,
      "b": 11,
      "P_b": 11,
      "fails_half": true
    },
    {
      "k": 5,
      "pow2": 32,
      "q": 17,
      "n": 49,
      "a": 32,
      "b": 17,
      "P_b": 17,
      "fails_half": true
    },
    {
      "k": 6,
      "pow2": 64,
      "q": 37,
      "n": 101,
      "a": 64,
      "b": 37,
      "P_b": 37,
      "fails_half": true
    },
    {
      "k": 7,
      "pow2": 128,
      "q": 67,
      "n": 195,
      "a": 128,
      "b": 67,
      "P_b": 67,
      "fails_half": true
    },
    {
      "k": 8,
      "pow2": 256,
      "q": 131,
      "n": 387,
      "a": 256,
      "b": 131,
      "P_b": 131,
      "fails_half": true
    },
    {
      "k": 9,
      "pow2": 512,
      "q": 257,
      "n": 769,
      "a": 512,
      "b": 257,
      "P_b": 257,
      "fails_half": true
    },
    {
      "k": 10,
      "pow2": 1024,
      "q": 521,
      "n": 1545,
      "a": 1024,
      "b": 521,
      "P_b": 521,
      "fails_half": true
    },
    {
      "k": 11,
      "pow2": 2048,
      "q": 1031,
      "n": 3079,
      "a": 2048,
      "b": 1031,
      "P_b": 1031,
      "fails_half": true
    },
    {
      "k": 12,
      "pow2": 4096,
      "q": 2053,
      "n": 6149,
      "a": 4096,
      "b": 2053,
      "P_b": 2053,
      "fails_half": true
    },
    {
      "k": 13,
      "pow2": 8192,
      "q": 4099,
      "n": 12291,
      "a": 8192,
      "b": 4099,
      "P_b": 4099,
      "fails_half": true
    },
    {
      "k": 14,
      "pow2": 16384,
      "q": 8209,
      "n": 24593,
      "a": 16384,
      "b": 8209,
      "P_b": 8209,
      "fails_half": true
    },
    {
      "k": 15,
      "pow2": 32768,
      "q": 16411,
      "n": 49179,
      "a": 32768,
      "b": 16411,
      "P_b": 16411,
      "fails_half": true
    },
    {
      "k": 16,
      "pow2": 65536,
      "q": 32771,
      "n": 98307,
      "a": 65536,
      "b": 32771,
      "P_b": 32771,
      "fails_half": true
    }
  ],
  "pow2_misses": [],
  "is_dent": false,
  "reason": "Explicit infinite-looking failure families for two closed-form templates. Not a lower bound on F; the square template still covers every n at exponent 1/2."
}