files / problems / two-smooth-summands / compute / q1 / certs / poly_obstruction.json
{
"statement": "A fixed polynomial of degree d>=2 produces remainder windows of exponent 1-1/d >= 1/2. Size-of-remainder covering therefore cannot beat the square template.",
"polys": [
{
"poly": "x^2",
"degree": 2,
"remainder_exponent": "1-1/2",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 400,
"gap": 41,
"n_at_max_remainder": 440,
"max_remainder": 40,
"gap_d": 1681,
"n_d_minus_1": 440,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 2500,
"gap": 101,
"n_at_max_remainder": 2600,
"max_remainder": 100,
"gap_d": 10201,
"n_d_minus_1": 2600,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 10000,
"gap": 201,
"n_at_max_remainder": 10200,
"max_remainder": 200,
"gap_d": 40401,
"n_d_minus_1": 10200,
"gap_beats_half_window": true
},
{
"k": 200,
"P_k": 40000,
"gap": 401,
"n_at_max_remainder": 40400,
"max_remainder": 400,
"gap_d": 160801,
"n_d_minus_1": 40400,
"gap_beats_half_window": true
},
{
"k": 400,
"P_k": 160000,
"gap": 801,
"n_at_max_remainder": 160800,
"max_remainder": 800,
"gap_d": 641601,
"n_d_minus_1": 160800,
"gap_beats_half_window": true
}
]
},
{
"poly": "x^3",
"degree": 3,
"remainder_exponent": "1-1/3",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 8000,
"gap": 1261,
"n_at_max_remainder": 9260,
"max_remainder": 1260,
"gap_d": 2005142581,
"n_d_minus_1": 85747600,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 125000,
"gap": 7651,
"n_at_max_remainder": 132650,
"max_remainder": 7650,
"gap_d": 447872715451,
"n_d_minus_1": 17596022500,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 1000000,
"gap": 30301,
"n_at_max_remainder": 1030300,
"max_remainder": 30300,
"gap_d": 27820881360901,
"n_d_minus_1": 1061518090000,
"gap_beats_half_window": true
},
{
"k": 200,
"P_k": 8000000,
"gap": 120601,
"n_at_max_remainder": 8120600,
"max_remainder": 120600,
"gap_d": 1754093449441801,
"n_d_minus_1": 65944144360000,
"gap_beats_half_window": true
},
{
"k": 400,
"P_k": 64000000,
"gap": 481201,
"n_at_max_remainder": 64481200,
"max_remainder": 481200,
"gap_d": 111424209989763601,
"n_d_minus_1": 4157825153440000,
"gap_beats_half_window": true
}
]
},
{
"poly": "x(x+1)",
"degree": 2,
"remainder_exponent": "1-1/2",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 420,
"gap": 42,
"n_at_max_remainder": 461,
"max_remainder": 41,
"gap_d": 1764,
"n_d_minus_1": 461,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 2550,
"gap": 102,
"n_at_max_remainder": 2651,
"max_remainder": 101,
"gap_d": 10404,
"n_d_minus_1": 2651,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 10100,
"gap": 202,
"n_at_max_remainder": 10301,
"max_remainder": 201,
"gap_d": 40804,
"n_d_minus_1": 10301,
"gap_beats_half_window": true
},
{
"k": 200,
"P_k": 40200,
"gap": 402,
"n_at_max_remainder": 40601,
"max_remainder": 401,
"gap_d": 161604,
"n_d_minus_1": 40601,
"gap_beats_half_window": true
},
{
"k": 400,
"P_k": 160400,
"gap": 802,
"n_at_max_remainder": 161201,
"max_remainder": 801,
"gap_d": 643204,
"n_d_minus_1": 161201,
"gap_beats_half_window": true
}
]
},
{
"poly": "x(x+1)/2",
"degree": 2,
"remainder_exponent": "1-1/2",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 210,
"gap": 21,
"n_at_max_remainder": 230,
"max_remainder": 20,
"gap_d": 441,
"n_d_minus_1": 230,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 1275,
"gap": 51,
"n_at_max_remainder": 1325,
"max_remainder": 50,
"gap_d": 2601,
"n_d_minus_1": 1325,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 5050,
"gap": 101,
"n_at_max_remainder": 5150,
"max_remainder": 100,
"gap_d": 10201,
"n_d_minus_1": 5150,
"gap_beats_half_window": true
},
{
"k": 200,
"P_k": 20100,
"gap": 201,
"n_at_max_remainder": 20300,
"max_remainder": 200,
"gap_d": 40401,
"n_d_minus_1": 20300,
"gap_beats_half_window": true
},
{
"k": 400,
"P_k": 80200,
"gap": 401,
"n_at_max_remainder": 80600,
"max_remainder": 400,
"gap_d": 160801,
"n_d_minus_1": 80600,
"gap_beats_half_window": true
}
]
},
{
"poly": "x^2+x+1",
"degree": 2,
"remainder_exponent": "1-1/2",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 421,
"gap": 42,
"n_at_max_remainder": 462,
"max_remainder": 41,
"gap_d": 1764,
"n_d_minus_1": 462,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 2551,
"gap": 102,
"n_at_max_remainder": 2652,
"max_remainder": 101,
"gap_d": 10404,
"n_d_minus_1": 2652,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 10101,
"gap": 202,
"n_at_max_remainder": 10302,
"max_remainder": 201,
"gap_d": 40804,
"n_d_minus_1": 10302,
"gap_beats_half_window": true
},
{
"k": 200,
"P_k": 40201,
"gap": 402,
"n_at_max_remainder": 40602,
"max_remainder": 401,
"gap_d": 161604,
"n_d_minus_1": 40602,
"gap_beats_half_window": true
},
{
"k": 400,
"P_k": 160401,
"gap": 802,
"n_at_max_remainder": 161202,
"max_remainder": 801,
"gap_d": 643204,
"n_d_minus_1": 161202,
"gap_beats_half_window": true
}
]
},
{
"poly": "x(x+1)(x+2)/6",
"degree": 3,
"remainder_exponent": "1-1/3",
"beats_square_by_size": false,
"samples": [
{
"k": 20,
"P_k": 1540,
"gap": 231,
"n_at_max_remainder": 1770,
"max_remainder": 230,
"gap_d": 12326391,
"n_d_minus_1": 3132900,
"gap_beats_half_window": true
},
{
"k": 50,
"P_k": 22100,
"gap": 1326,
"n_at_max_remainder": 23425,
"max_remainder": 1325,
"gap_d": 2331473976,
"n_d_minus_1": 548730625,
"gap_beats_half_window": true
},
{
"k": 100,
"P_k": 171700,
"gap": 5151,
"n_at_max_remainder": 176850,
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"gap_d": 136670457951,
"n_d_minus_1": 31275922500,
"gap_beats_half_window": true
},
{
"k": 200,
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"n_at_max_remainder": 1373700,
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"gap_d": 8366663330901,
"n_d_minus_1": 1887051690000,
"gap_beats_half_window": true
},
{
"k": 400,
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"n_d_minus_1": 117232590760000,
"gap_beats_half_window": true
}
]
}
],
"is_dent": false,
"size_lower_bound_holds": true
}