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11193 bytes, as of the version from 2026-09-16 17:58 (current). Recorded here, not run.
"""Good examples of Hall's conjecture -- numberdb.org/T278.
k = y^2 - x^3, r = sqrt(x) / |k|
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
The table stores the 54 examples listed by Wikipedia's Hall's conjecture page:
the integer x, the nearest integer y to x^(3/2), the signed Mordell constant
k = y^2 - x^3, and the Hall ratio r. Wikipedia gives x and a rounded r; y and
k are recomputed exactly from x. Elkies' independent list for x < 10^18 uses
the opposite sign for k, and is checked here after changing that sign.
"""
import os
import sys
from math import isqrt
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField
TABLE = os.environ.get("NUMBERDB_TABLE", "T278")
DIGITS = 30
WORKING_GUARD = 80
HALL_DATA = (
(1, 2, "1.41"),
(2, 5234, "4.26"),
(3, 8158, "3.76"),
(4, 93844, "1.03"),
(5, 367806, "2.93"),
(6, 421351, "1.05"),
(7, 720114, "3.77"),
(8, 939787, "3.16"),
(9, 28187351, "4.87"),
(10, 110781386, "1.23"),
(11, 154319269, "1.08"),
(12, 384242766, "1.34"),
(13, 390620082, "1.33"),
(14, 3790689201, "2.20"),
(15, 65589428378, "2.19"),
(16, 952764389446, "1.15"),
(17, 12438517260105, "1.27"),
(18, 35495694227489, "1.15"),
(19, 53197086958290, "1.66"),
(20, 5853886516781223, "46.60"),
(21, 12813608766102806, "1.30"),
(22, 23415546067124892, "1.46"),
(23, 38115991067861271, "6.50"),
(24, 322001299796379844, "1.04"),
(25, 471477085999389882, "1.38"),
(26, 810574762403977064, "4.66"),
(27, 9870884617163518770, "1.90"),
(28, 42532374580189966073, "3.47"),
(29, 44648329463517920535, "1.79"),
(30, 51698891432429706382, "1.75"),
(31, 231411667627225650649, "3.71"),
(32, 601724682280310364065, "1.88"),
(33, 4996798823245299750533, "2.17"),
(34, 5592930378182848874404, "1.38"),
(35, 14038790674256691230847, "1.27"),
(36, 77148032713960680268604, "10.18"),
(37, 180179004295105849668818, "5.65"),
(38, 372193377967238474960883, "1.33"),
(39, 664947779818324205678136, "16.53"),
(40, 2028871373185892500636155, "1.14"),
(41, 10747835083471081268825856, "1.35"),
(42, 37223900078734215181946587, "1.87"),
(43, 69586951610485633367491417, "1.22"),
(44, 3690445383173227306376634720, "1.51"),
(45, 133545763574262054617147641349, "1.69"),
(46, 162921297743817207342396140787, "10.65"),
(47, 374192690896219210878121645171, "2.97"),
(48, 401844774500818781164623821177, "1.29"),
(49, 500859224588646106403669009291, "1.06"),
(50, 1114592308630995805123571151844, "1.04"),
(51, 39739590925054773507790363346813, "3.75"),
(52, 862611143810724763613366116643858, "1.10"),
(53, 1062521751024771376590062279975859, "1.006"),
(54, 6078673043126084065007902175846955, "1.03"),
)
# Elkies lists k = x^3 - y^2, opposite to this table's convention.
ELKIES_DATA = (
(5853886516781223, 1641843),
(38115991067861271, 30032270),
(28187351, -1090),
(810574762403977064, -193234265),
(5234, -17),
(720114, -225),
(8158, -24),
(939787, 307),
(367806, 207),
(3790689201, -28024),
(65589428378, -117073),
(53197086958290, -4401169),
(23415546067124892, 105077952),
(2, -1),
(471477085999389882, -497218657),
(384242766, -14668),
(390620082, -14857),
(12813608766102806, -87002345),
(12438517260105, 2767769),
(110781386, -8569),
(35495694227489, 5190544),
(154319269, -11492),
(421351, -618),
(322001299796379844, 548147655),
(93844, -297),
)
_CHECKED = False
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def _nearest_y(x):
"""The nearest integer to x^(3/2), using integer arithmetic only."""
cube = x ** 3
y = isqrt(cube)
if (y + 1) ** 2 - cube < cube - y ** 2:
y += 1
return ZZ(y)
def _hall_parts(x):
x = ZZ(x)
y = _nearest_y(int(x))
k = y ** 2 - x ** 3
return x, y, ZZ(k)
def _source_ratio_parts(text):
whole, _, fractional = text.partition(".")
scale = 10 ** len(fractional)
return int(whole + fractional), scale
def _ratio_rounds_to_source(x, k, source):
"""Check the source's rounded r without computing decimal digits."""
numerator, scale = _source_ratio_parts(source)
left = (2 * numerator - 1) ** 2 * int(k) ** 2
middle = 4 * scale ** 2 * int(x)
right = (2 * numerator + 1) ** 2 * int(k) ** 2
return left <= middle < right
def _chowla_t3():
"""The Birch-Chowla-Hall-Schinzel family at t = 3."""
t = ZZ(3)
x = t * (t ** 9 + 6 * t ** 6 + 15 * t ** 3 + 12) // ZZ(9)
k = (3 * t ** 6 + 14 * t ** 3 + 27) // ZZ(108)
return x, k
def _check_data():
"""Exact checks against the source ratios and Elkies' independent list."""
global _CHECKED
if _CHECKED:
return
if len(HALL_DATA) != 54:
raise ArithmeticError("expected 54 Hall examples")
if [index for index, _x, _ratio in HALL_DATA] != list(range(1, 55)):
raise ArithmeticError("the source row numbers are not consecutive")
by_x = {}
for index, x, source_ratio in HALL_DATA:
if x in by_x:
raise ArithmeticError("duplicate x in source list: %s" % x)
by_x[x] = index
_x, _y, k = _hall_parts(x)
if k == 0:
raise ArithmeticError("row %s has k = 0" % index)
if not _ratio_rounds_to_source(x, k, source_ratio):
raise ArithmeticError(
"row %s with x = %s does not round to source ratio %s" %
(index, x, source_ratio))
row20 = _hall_parts(5853886516781223)
if row20[1] != 447884928428402042307918 or row20[2] != -1641843:
raise ArithmeticError("Elkies record row does not reproduce")
row22 = _hall_parts(23415546067124892)
if row22[2] != 64 * row20[2]:
raise ArithmeticError("the non-primitive row does not scale k by 64")
chowla_x, chowla_k = _chowla_t3()
if _hall_parts(chowla_x)[2] != chowla_k:
raise ArithmeticError("Chowla's t = 3 example does not reproduce row 3")
# Wikipedia says Elkies' own table omits its entry 16.
omitted_by_elkies = 952764389446
wikipedia_under_bound = {
x for _index, x, _ratio in HALL_DATA
if x < 10 ** 18 and x != omitted_by_elkies
}
elkies_x = {x for x, _k in ELKIES_DATA}
if wikipedia_under_bound != elkies_x:
raise ArithmeticError(
"Wikipedia and Elkies list different x < 10^18 values: %s / %s" %
(sorted(wikipedia_under_bound - elkies_x),
sorted(elkies_x - wikipedia_under_bound)))
for x, elkies_k in ELKIES_DATA:
if _hall_parts(x)[2] != -ZZ(elkies_k):
raise ArithmeticError(
"Elkies gives k = %s for x = %s, but this convention gives %s" %
(elkies_k, x, _hall_parts(x)[2]))
_CHECKED = True
def _ratio(x, k, digits):
field = RealBallField(numberdb.bits(digits, losing=WORKING_GUARD))
return field(x).sqrt() / field(abs(k))
def _entry_comment(x, part):
if x == 2 and part == "r":
return "Here $r=\\sqrt{2}$."
if x == 5853886516781223 and part == "x":
return "This row has the largest Hall ratio in the source list."
if x == 23415546067124892 and part == "x":
return (
"This non-primitive example is obtained from the Elkies row "
"$x=5853886516781223$ by multiplying $x$, $y$ and $k$ by "
"$4$, $8$ and $64$."
)
return None
class GoodHallExamples(numberdb.Generator):
table = TABLE
parameters = ("x", "part")
type = "R"
digits = DIGITS
rigour = "proven"
def enumerate(self):
_check_data()
for _index, x, _source_ratio in HALL_DATA:
for part in ("x", "y", "k", "r"):
yield {"x": str(x), "part": part}
def value(self, params, digits):
x, y, k = _hall_parts(int(params["x"]))
part = params["part"]
if part == "x":
number = x
elif part == "y":
number = y
elif part == "k":
number = k
elif part == "r":
source = next(ratio for _i, listed_x, ratio in HALL_DATA
if listed_x == int(x))
if not _ratio_rounds_to_source(x, k, source):
raise ArithmeticError("r for x = %s no longer matches %s" %
(x, source))
number = _ratio(x, k, digits)
else:
raise ValueError("unknown part %s" % part)
comment = _entry_comment(int(x), part)
if comment:
return {"number": number, "comment": comment}
return number
def fill_draft_once(generator, message):
"""Fill a fresh draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message,
rigour=generator.rigour)
return answer
if __name__ == "__main__":
_key_from_stdin()
generator = GoodHallExamples()
_check_data()
if "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1":
print(fill_draft_once(
generator,
message="Hall examples from the Wikipedia list, with y and k "
"recomputed exactly from x and r in ball arithmetic"))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)