files / problems / hilbert16-limit-cycles / compute / q2 / o-iterated-squaring / certs / arithmetic.json

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{
  "n": 3,
  "N_formula": "(n+1) 2^k - 1",
  "rows": [
    {
      "k": 1,
      "N": 7,
      "sheet_ceiling_bezout": 4,
      "actual_complex": 2,
      "chebyshev_m": 2,
      "chebyshev_sheets": 4,
      "bezout_over_N2": [
        4,
        49
      ],
      "complex_over_N2": [
        2,
        49
      ],
      "complex_eq_(N+1)/(n+1)": true
    },
    {
      "k": 2,
      "N": 15,
      "sheet_ceiling_bezout": 16,
      "actual_complex": 4,
      "chebyshev_m": 4,
      "chebyshev_sheets": 16,
      "bezout_over_N2": [
        16,
        225
      ],
      "complex_over_N2": [
        4,
        225
      ],
      "complex_eq_(N+1)/(n+1)": true
    },
    {
      "k": 3,
      "N": 31,
      "sheet_ceiling_bezout": 64,
      "actual_complex": 8,
      "chebyshev_m": 8,
      "chebyshev_sheets": 64,
      "bezout_over_N2": [
        64,
        961
      ],
      "complex_over_N2": [
        8,
        961
      ],
      "complex_eq_(N+1)/(n+1)": true
    },
    {
      "k": 4,
      "N": 63,
      "sheet_ceiling_bezout": 256,
      "actual_complex": 16,
      "chebyshev_m": 16,
      "chebyshev_sheets": 256,
      "bezout_over_N2": [
        256,
        3969
      ],
      "complex_over_N2": [
        16,
        3969
      ],
      "complex_eq_(N+1)/(n+1)": true
    },
    {
      "k": 5,
      "N": 127,
      "sheet_ceiling_bezout": 1024,
      "actual_complex": 32,
      "chebyshev_m": 32,
      "chebyshev_sheets": 1024,
      "bezout_over_N2": [
        1024,
        16129
      ],
      "complex_over_N2": [
        32,
        16129
      ],
      "complex_eq_(N+1)/(n+1)": true
    },
    {
      "k": 6,
      "N": 255,
      "sheet_ceiling_bezout": 4096,
      "actual_complex": 64,
      "chebyshev_m": 64,
      "chebyshev_sheets": 4096,
      "bezout_over_N2": [
        4096,
        65025
      ],
      "complex_over_N2": [
        64,
        65025
      ],
      "complex_eq_(N+1)/(n+1)": true
    }
  ],
  "conclusion": {
    "beats_theorem2": false,
    "attains_m2_per_step": false,
    "actual_complex_growth": "linear",
    "complex_sheets_formula": "(N+1)/(n+1)",
    "bezout_ceiling_growth": "quadratic",
    "answers_remark4": "negative_for_complex_squaring",
    "do_not_claim_252_1080_1380_2012": true
  }
}