text stringlengths 34 210 |
|---|
0 = (-5)^3 + (6)^3 + (-4)^3 + (-3)^3 |
1 = (-6)^3 + (7)^3 + (-1)^3 + (-5)^3 |
2 = (-3)^3 + (4)^3 + (-3)^3 + (-2)^3 |
3 = (235)^3 + (-234)^3 + (-56)^3 + (22)^3 |
4 = (4)^3 + (-5)^3 + (4)^3 + (1)^3 |
5 = (2)^3 + (-1)^3 + (-1)^3 + (-1)^3 |
6 = (-58)^3 + (59)^3 + (-21)^3 + (-10)^3 |
7 = (5)^3 + (-4)^3 + (-3)^3 + (-3)^3 |
8 = (14)^3 + (-13)^3 + (-8)^3 + (-3)^3 |
9 = (-108)^3 + (109)^3 + (-27)^3 + (-25)^3 |
10 = (-2)^3 + (3)^3 + (-2)^3 + (-1)^3 |
11 = (-2)^3 + (3)^3 + (-2)^3 + (0)^3 |
12 = (-186)^3 + (187)^3 + (-47)^3 + (-8)^3 |
13 = (10)^3 + (-11)^3 + (7)^3 + (1)^3 |
14 = (-235)^3 + (236)^3 + (2)^3 + (-55)^3 |
15 = (842)^3 + (-841)^3 + (-132)^3 + (56)^3 |
16 = (-19)^3 + (20)^3 + (-10)^3 + (-5)^3 |
17 = (3)^3 + (-2)^3 + (-1)^3 + (-1)^3 |
18 = (-95)^3 + (96)^3 + (-7)^3 + (-30)^3 |
19 = (-5)^3 + (6)^3 + (-4)^3 + (-2)^3 |
20 = (-20)^3 + (21)^3 + (-9)^3 + (-8)^3 |
21 = (10)^3 + (-9)^3 + (-5)^3 + (-5)^3 |
22 = (85)^3 + (-86)^3 + (28)^3 + (1)^3 |
23 = (146)^3 + (-145)^3 + (-40)^3 + (8)^3 |
24 = (-16)^3 + (17)^3 + (-9)^3 + (-4)^3 |
25 = (-136)^3 + (137)^3 + (-38)^3 + (-10)^3 |
26 = (-4)^3 + (5)^3 + (-3)^3 + (-2)^3 |
27 = (-11)^3 + (12)^3 + (-7)^3 + (-3)^3 |
28 = (-3)^3 + (4)^3 + (-2)^3 + (-1)^3 |
29 = (-3)^3 + (4)^3 + (-2)^3 + (0)^3 |
30 = (120)^3 + (-119)^3 + (-35)^3 + (4)^3 |
31 = (229687)^3 + (-229688)^3 + (7390)^3 + (-6260)^3 |
32 = (431)^3 + (-430)^3 + (-94)^3 + (65)^3 |
33 = (59)^3 + (-58)^3 + (-19)^3 + (-15)^3 |
34 = (-4)^3 + (5)^3 + (-3)^3 + (0)^3 |
35 = (4)^3 + (-3)^3 + (-1)^3 + (-1)^3 |
36 = (-6)^3 + (7)^3 + (-4)^3 + (-3)^3 |
37 = (6)^3 + (-5)^3 + (-3)^3 + (-3)^3 |
38 = (-41)^3 + (42)^3 + (-17)^3 + (-6)^3 |
39 = (117)^3 + (-116)^3 + (-35)^3 + (13)^3 |
40 = (148)^3 + (-149)^3 + (40)^3 + (13)^3 |
41 = (8)^3 + (-7)^3 + (-4)^3 + (-4)^3 |
42 = (-109)^3 + (110)^3 + (15)^3 + (-34)^3 |
43 = (-7)^3 + (8)^3 + (-5)^3 + (-1)^3 |
44 = (-7)^3 + (8)^3 + (-5)^3 + (0)^3 |
45 = (-44)^3 + (45)^3 + (-18)^3 + (-4)^3 |
46 = (25)^3 + (-24)^3 + (-3)^3 + (-12)^3 |
47 = (-9)^3 + (10)^3 + (-6)^3 + (-2)^3 |
48 = (-21)^3 + (22)^3 + (-11)^3 + (-2)^3 |
49 = (-2)^3 + (1)^3 + (4)^3 + (-2)^3 |
50 = (-76)^3 + (77)^3 + (-22)^3 + (-19)^3 |
51 = (-10)^3 + (11)^3 + (-6)^3 + (-4)^3 |
52 = (-4)^3 + (5)^3 + (-2)^3 + (-1)^3 |
53 = (-4)^3 + (5)^3 + (-2)^3 + (0)^3 |
54 = (-9)^3 + (10)^3 + (-6)^3 + (-1)^3 |
55 = (-6)^3 + (7)^3 + (-4)^3 + (-2)^3 |
56 = (-5)^3 + (6)^3 + (-3)^3 + (-2)^3 |
57 = (-21)^3 + (22)^3 + (1)^3 + (-11)^3 |
58 = (1)^3 + (-2)^3 + (4)^3 + (1)^3 |
59 = (5)^3 + (-4)^3 + (-1)^3 + (-1)^3 |
60 = (1202)^3 + (-1201)^3 + (-163)^3 + (0)^3 |
61 = (-16)^3 + (17)^3 + (-9)^3 + (-3)^3 |
62 = (7)^3 + (-6)^3 + (-4)^3 + (-1)^3 |
63 = (-30)^3 + (31)^3 + (-12)^3 + (-10)^3 |
64 = (-5)^3 + (6)^3 + (-3)^3 + (0)^3 |
65 = (-5)^3 + (6)^3 + (-3)^3 + (1)^3 |
66 = (-198)^3 + (199)^3 + (-50)^3 + (19)^3 |
67 = (-5)^3 + (4)^3 + (4)^3 + (4)^3 |
68 = (257)^3 + (-256)^3 + (-58)^3 + (-13)^3 |
69 = (-148)^3 + (149)^3 + (-42)^3 + (20)^3 |
70 = (-58)^3 + (59)^3 + (-20)^3 + (-13)^3 |
71 = (-6)^3 + (7)^3 + (2)^3 + (-4)^3 |
72 = (-14)^3 + (15)^3 + (-7)^3 + (-6)^3 |
73 = (7)^3 + (-6)^3 + (-3)^3 + (-3)^3 |
74 = (25)^3 + (-24)^3 + (-12)^3 + (1)^3 |
75 = (-53)^3 + (54)^3 + (-20)^3 + (-8)^3 |
76 = (10)^3 + (-11)^3 + (7)^3 + (4)^3 |
77 = (-19)^3 + (20)^3 + (-10)^3 + (-4)^3 |
78 = (-7)^3 + (8)^3 + (-4)^3 + (-3)^3 |
79 = (-13)^3 + (14)^3 + (-7)^3 + (-5)^3 |
80 = (-16)^3 + (17)^3 + (-9)^3 + (-2)^3 |
81 = (11)^3 + (-10)^3 + (-5)^3 + (-5)^3 |
82 = (-5)^3 + (6)^3 + (-2)^3 + (-1)^3 |
83 = (-5)^3 + (6)^3 + (-2)^3 + (0)^3 |
84 = (-8)^3 + (9)^3 + (-5)^3 + (-2)^3 |
85 = (-86)^3 + (85)^3 + (28)^3 + (4)^3 |
86 = (-106795)^3 + (106796)^3 + (-4039)^3 + (3164)^3 |
87 = (-16)^3 + (17)^3 + (-1)^3 + (-9)^3 |
88 = (473)^3 + (-472)^3 + (-90)^3 + (39)^3 |
89 = (6)^3 + (-5)^3 + (-1)^3 + (-1)^3 |
90 = (-188)^3 + (189)^3 + (-43)^3 + (-30)^3 |
91 = (9)^3 + (-8)^3 + (-5)^3 + (-1)^3 |
92 = (-6)^3 + (7)^3 + (-3)^3 + (-2)^3 |
93 = (165)^3 + (-164)^3 + (-44)^3 + (16)^3 |
94 = (535)^3 + (-536)^3 + (94)^3 + (31)^3 |
95 = (-130)^3 + (131)^3 + (-7)^3 + (-37)^3 |
96 = (17)^3 + (-16)^3 + (-9)^3 + (2)^3 |
97 = (-7)^3 + (8)^3 + (-4)^3 + (-2)^3 |
98 = (26)^3 + (-25)^3 + (-12)^3 + (-5)^3 |
99 = (13)^3 + (-12)^3 + (-7)^3 + (-3)^3 |
End of preview. Expand in Data Studio
Sum of Four Cubes Certificates
This dataset contains explicit certificates for
n = x1^3 + x2^3 + x3^3 + x4^3
for every integer 0 <= n <= 1,000,000,000.
Each row has the form
123 = (x1)^3 + (x2)^3 + (x3)^3 + (x4)^3
The files are sharded by intervals of length 10,000,000, with the endpoint
1,000,000,000 appended to the last shard:
cubes-00.txt 0 <= n <= 9,999,999
cubes-01.txt 10,000,000 <= n <= 19,999,999
...
cubes-99.txt 990,000,000 <= n <= 1,000,000,000
The generation code uses a congruence-filtered generalized Pell search. For
the first successful search parameter d, it records the representation with
minimal height
H = max(|x1|, |x2|, |x3|, |x4|)
among the parity-compatible Pell representatives returned by the reference solver.
Reproduce
Generate shards with 15 worker processes:
sage -python scripts/generate_hf_dataset.py \
--start 0 --end 1000000000 \
--workers 15 \
--chunk-size 10000 \
--max-in-flight 30 \
--out-dir generated/sum4cubes \
--overwrite \
--merge-final-singleton \
--progress-every 10000000
For long runs, restart the generator from shard boundaries with --resume.
For example:
sage -python scripts/generate_hf_dataset.py \
--start 80000000 --end 129999999 \
--workers 15 \
--chunk-size 10000 \
--max-in-flight 30 \
--out-dir generated/sum4cubes \
--resume
Run a lightweight post-generation sanity check:
python3 scripts/check_generated_shards.py \
--dir generated/sum4cubes \
--start 0 --end 1000000000
The generator verifies each identity using exact integer arithmetic before writing a row. The sanity checker verifies shard coverage and exact four-cube identities for boundary rows.
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