File size: 2,809 Bytes
b8f9934 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | """Reference Sage/Python implementation for the four-cubes Pell search.
Run with:
sage -python scripts/four_cubes_reference.py
or pass targets explicitly:
sage -python scripts/four_cubes_reference.py 254 314159265358979323
"""
import sys
from sage.all import ZZ, BinaryQF, Mod, factor, is_square
def local_admissible(n, d, M):
"""Return whether the local square tests from Lemma 3.3 pass."""
for p, a in factor(d):
q = p**a
if not Mod(M, q).is_square():
return False
return True
def solve_pell_with_congruences(d, M):
"""Find Pell candidates satisfying the required parity conditions."""
Q = BinaryQF([1, 0, -d])
for u, v in Q.solve_integer(M, _flag=3):
u, v = ZZ(u), ZZ(v)
if (u - 1) % 2 == 0 and (v - d) % 2 == 0:
yield u, v
def quadruple_from_pell(d, u, v):
"""Convert a parity-compatible Pell solution to a four-cube tuple."""
x = (u - 1) // 2
y = (v - d) // 2
return (x + 1, -x, -(y + d), y)
def height(xs):
"""Return H=max_i |x_i|."""
return max(abs(x) for x in xs)
def solve(n, limit=10**8, return_d=False):
"""Return one quadruple whose cubes sum to n, or None."""
n = ZZ(n)
if n % 9 == 4:
sol = solve(-n, limit=limit, return_d=True)
if sol is None:
return None
x1, x2, x3, x4, d = sol
out = (-x1, -x2, -x3, -x4)
return out + (d,) if return_d else out
residue = ZZ((1 - n) % 6)
if residue == 0:
residue = ZZ(6)
for d in range(residue, limit + 1, 6):
d = ZZ(d)
if is_square(d):
continue
numerator = 4*n - 1 + d**3
if numerator % 3 != 0:
continue
M = numerator // 3
if not local_admissible(n, d, M):
continue
candidates = [
quadruple_from_pell(d, u, v)
for u, v in solve_pell_with_congruences(d, M)
]
if not candidates:
continue
out = min(
candidates,
key=lambda xs: (height(xs), tuple(xs)),
)
return out + (d,) if return_d else out
return None
def verify(n, xs):
"""Check x1^3 + x2^3 + x3^3 + x4^3 = n exactly."""
return sum(ZZ(x)**3 for x in xs) == ZZ(n)
def main(argv):
targets = [ZZ(arg) for arg in argv]
if not targets:
targets = [
ZZ(254),
ZZ(314159265358979323),
ZZ(27182818284590452),
]
for n in targets:
sol = solve(n, limit=2000, return_d=True)
ok = sol is not None and verify(n, sol[:4])
print(f"n = {n}")
print(f"solution = {sol}")
print(f"verified = {ok}")
if not ok:
raise SystemExit(1)
if __name__ == "__main__":
main(sys.argv[1:])
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