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Gro-Tsen/stoermers-problem.sage

Created February 17, 2022 22:36
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stoermers-problem.sage
# Return cnt solutions of the Pell-Fermat equation x^2 - d*y^2 = 1
def pell_equation(d, cnt):
if cnt < 1:
return []
# First, find the "fundamental" solution
cf = continued_fraction(QQbar(sqrt(d)))
i = 0
(xfun,yfun) = (None,None)
while True:
r = cf.convergent(i)
(x,y) = (r.numerator(), r.denominator())
if x^2 - d*y^2 == 1:
(xfun,yfun) = (x,y)
break
i += 1
res = [(xfun,yfun)]
(x,y) = (xfun, yfun)
while len(res) < cnt:
(x,y) = (x*xfun + d*y*yfun, x*yfun + y*xfun)
res.append((x,y))
return res
# Solve Størmer's problem (finding all pairs of consecutive integers
# that are smooth in the sense that their factors are within the given
# set), using Lehmer's method:
def lehmer_method(primes):
primeset = set(primes)
primelist = list(primes)
res = []
num = len(primelist)
largest = max(primelist)
bound = max(3,(largest+1)/2)
# Consider every Pell equation for 2*q where q is a product of a
# subset of the given primes:
for i in range(2^num):
q = prod([primelist[j] for j in range(num) if (i>>j)&1==1])
if q == 2:
continue
print("%d/%d:"%(i,2^num), "solving Pell's equation for", q)
pell = pell_equation(2*q, bound)
for (x,y) in pell:
if x<=1:
continue
u = ZZ((x-1)/2)
v = ZZ((x+1)/2)
# Check whether u and v are smooth:
good = True
for w in [u,v]:
for p in primelist:
while ZZ(p).divides(w):
w /= p
if w>1:
good = False
break
if good:
res.append((u,v))
res.sort(key=lambda p: p[0])
return res
# The set of supersingular primes (to be used as an example):
supersingulars = [nth_prime(i+1) for i in range(11)] + [41,47,59,71]
sssmooth = lehmer_method(supersingulars)
# Result (after a long computation) is (len==1310):
# sssmooth = [
# (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9),
# (9, 10), (10, 11), (11, 12), (12, 13), (13, 14), (14, 15),
# (15, 16), (16, 17), (17, 18), (18, 19), (19, 20), (20, 21),
# (21, 22), (22, 23), (23, 24), (24, 25), (25, 26), (26, 27),
# (27, 28), (28, 29), (29, 30), (30, 31), (31, 32), (32, 33),
# (33, 34), (34, 35), (35, 36), (38, 39), (39, 40), (40, 41),
# (41, 42), (44, 45), (45, 46), (46, 47), (47, 48), (48, 49),
# (49, 50), (50, 51), (51, 52), (54, 55), (55, 56), (56, 57),
# (57, 58), (58, 59), (59, 60), (62, 63), (63, 64), (64, 65),
# (65, 66), (68, 69), (69, 70), (70, 71), (71, 72), (75, 76),
# (76, 77), (77, 78), (80, 81), (81, 82), (84, 85), (87, 88),
# (90, 91), (91, 92), (92, 93), (93, 94), (94, 95), (95, 96),
# (98, 99), (99, 100), (104, 105), (114, 115), (115, 116),
# (116, 117), (117, 118), (118, 119), (119, 120), (120, 121),
# (123, 124), (124, 125), (125, 126), (132, 133), (135, 136),
# (140, 141), (141, 142), (142, 143), (143, 144), (144, 145),
# (152, 153), (153, 154), (154, 155), (155, 156), (160, 161),
# (161, 162), (164, 165), (168, 169), (169, 170), (170, 171),
# (174, 175), (175, 176), (176, 177), (186, 187), (187, 188),
# (188, 189), (189, 190), (195, 196), (203, 204), (204, 205),
# (207, 208), (208, 209), (209, 210), (216, 217), (220, 221),
# (224, 225), (230, 231), (231, 232), (234, 235), (235, 236),
# (242, 243), (245, 246), (246, 247), (247, 248), (252, 253),
# (255, 256), (260, 261), (272, 273), (275, 276), (279, 280),
# (284, 285), (285, 286), (286, 287), (287, 288), (288, 289),
# (289, 290), (294, 295), (299, 300), (319, 320), (322, 323),
# (323, 324), (324, 325), (328, 329), (329, 330), (340, 341),
# (341, 342), (342, 343), (350, 351), (351, 352), (354, 355),
# (360, 361), (363, 364), (368, 369), (374, 375), (375, 376),
# (376, 377), (377, 378), (384, 385), (390, 391), (391, 392),
# (399, 400), (405, 406), (413, 414), (425, 426), (434, 435),
# (440, 441), (441, 442), (450, 451), (455, 456), (459, 460),
# (464, 465), (475, 476), (483, 484), (492, 493), (493, 494),
# (494, 495), (495, 496), (496, 497), (506, 507), (512, 513),
# (527, 528), (528, 529), (531, 532), (532, 533), (539, 540),
# (550, 551), (551, 552), (560, 561), (567, 568), (574, 575),
# (575, 576), (588, 589), (589, 590), (594, 595), (608, 609),
# (611, 612), (615, 616), (620, 621), (624, 625), (637, 638),
# (638, 639), (639, 640), (648, 649), (649, 650), (650, 651),
# (675, 676), (696, 697), (704, 705), (713, 714), (714, 715),
# (725, 726), (728, 729), (735, 736), (759, 760), (767, 768),
# (779, 780), (780, 781), (781, 782), (782, 783), (783, 784),
# (798, 799), (799, 800), (805, 806), (819, 820), (825, 826),
# (832, 833), (836, 837), (840, 841), (845, 846), (846, 847),
# (867, 868), (874, 875), (884, 885), (896, 897), (899, 900),
# (923, 924), (930, 931), (935, 936), (943, 944), (944, 945),
# (960, 961), (968, 969), (986, 987), (987, 988), (1000, 1001),
# (1014, 1015), (1023, 1024), (1024, 1025), (1025, 1026),
# (1034, 1035), (1044, 1045), (1053, 1054), (1064, 1065),
# (1065, 1066), (1080, 1081), (1088, 1089), (1104, 1105),
# (1120, 1121), (1121, 1122), (1127, 1128), (1155, 1156),
# (1175, 1176), (1188, 1189), (1189, 1190), (1196, 1197),
# (1209, 1210), (1215, 1216), (1224, 1225), (1239, 1240),
# (1274, 1275), (1275, 1276), (1287, 1288), (1311, 1312),
# (1330, 1331), (1349, 1350), (1352, 1353), (1363, 1364),
# (1364, 1365), (1394, 1395), (1420, 1421), (1425, 1426),
# (1444, 1445), (1449, 1450), (1456, 1457), (1457, 1458),
# (1475, 1476), (1495, 1496), (1518, 1519), (1519, 1520),
# (1520, 1521), (1539, 1540), (1550, 1551), (1595, 1596),
# (1598, 1599), (1599, 1600), (1624, 1625), (1632, 1633),
# (1652, 1653), (1680, 1681), (1681, 1682), (1682, 1683),
# (1700, 1701), (1704, 1705), (1710, 1711), (1715, 1716),
# (1728, 1729), (1767, 1768), (1770, 1771), (1785, 1786),
# (1804, 1805), (1845, 1846), (1859, 1860), (1862, 1863),
# (1880, 1881), (1885, 1886), (1988, 1989), (2000, 2001),
# (2001, 2002), (2015, 2016), (2023, 2024), (2024, 2025),
# (2057, 2058), (2058, 2059), (2079, 2080), (2090, 2091),
# (2115, 2116), (2124, 2125), (2175, 2176), (2184, 2185),
# (2200, 2201), (2204, 2205), (2208, 2209), (2209, 2210),
# (2232, 2233), (2254, 2255), (2255, 2256), (2261, 2262),
# (2295, 2296), (2299, 2300), (2300, 2301), (2303, 2304),
# (2349, 2350), (2375, 2376), (2400, 2401), (2414, 2415),
# (2418, 2419), (2419, 2420), (2430, 2431), (2431, 2432),
# (2464, 2465), (2484, 2485), (2499, 2500), (2541, 2542),
# (2583, 2584), (2584, 2585), (2600, 2601), (2624, 2625),
# (2639, 2640), (2645, 2646), (2697, 2698), (2736, 2737),
# (2754, 2755), (2772, 2773), (2783, 2784), (2820, 2821),
# (2870, 2871), (2890, 2891), (2911, 2912), (2925, 2926),
# (2944, 2945), (2975, 2976), (3008, 3009), (3024, 3025),
# (3059, 3060), (3068, 3069), (3124, 3125), (3135, 3136),
# (3185, 3186), (3195, 3196), (3248, 3249), (3249, 3250),
# (3266, 3267), (3289, 3290), (3362, 3363), (3363, 3364),
# (3380, 3381), (3450, 3451), (3479, 3480), (3480, 3481),
# (3509, 3510), (3519, 3520), (3549, 3550), (3564, 3565),
# (3689, 3690), (3717, 3718), (3750, 3751), (3772, 3773),
# (3807, 3808), (3834, 3835), (3875, 3876), (3887, 3888),
# (3894, 3895), (3905, 3906), (3968, 3969), (4046, 4047),
# (4047, 4048), (4059, 4060), (4095, 4096), (4130, 4131),
# (4185, 4186), (4199, 4200), (4224, 4225), (4263, 4264),
# (4277, 4278), (4370, 4371), (4374, 4375), (4464, 4465),
# (4465, 4466), (4484, 4485), (4543, 4544), (4640, 4641),
# (4674, 4675), (4692, 4693), (4719, 4720), (4760, 4761),
# (4784, 4785), (4899, 4900), (4900, 4901), (4913, 4914),
# (4959, 4960), (4960, 4961), (4991, 4992), (5015, 5016),
# (5040, 5041), (5075, 5076), (5082, 5083), (5083, 5084),
# (5103, 5104), (5264, 5265), (5324, 5325), (5452, 5453),
# (5487, 5488), (5499, 5500), (5576, 5577), (5642, 5643),
# (5680, 5681), (5750, 5751), (5775, 5776), (5780, 5781),
# (5781, 5782), (5796, 5797), (5831, 5832), (5887, 5888),
# (5915, 5916), (5928, 5929), (5984, 5985), (6075, 6076),
# (6136, 6137), (6137, 6138), (6174, 6175), (6250, 6251),
# (6272, 6273), (6292, 6293), (6324, 6325), (6460, 6461),
# (6479, 6480), (6560, 6561), (6600, 6601), (6655, 6656),
# (6669, 6670), (6720, 6721), (6726, 6727), (6727, 6728),
# (6785, 6786), (6815, 6816), (6859, 6860), (6902, 6903),
# (6929, 6930), (7020, 7021), (7105, 7106), (7139, 7140),
# (7175, 7176), (7191, 7192), (7315, 7316), (7424, 7425),
# (7656, 7657), (7667, 7668), (7749, 7750), (7865, 7866),
# (7888, 7889), (7904, 7905), (7935, 7936), (7942, 7943),
# (8036, 8037), (8091, 8092), (8280, 8281), (8319, 8320),
# (8378, 8379), (8448, 8449), (8449, 8450), (8463, 8464),
# (8525, 8526), (8554, 8555), (8555, 8556), (8568, 8569),
# (8624, 8625), (8648, 8649), (8670, 8671), (8788, 8789),
# (8835, 8836), (8855, 8856), (8874, 8875), (8959, 8960),
# (9024, 9025), (9424, 9425), (9499, 9500), (9728, 9729),
# (9775, 9776), (9800, 9801), (9801, 9802), (10010, 10011),
# (10044, 10045), (10152, 10153), (10199, 10200), (10240, 10241),
# (10295, 10296), (10373, 10374), (10527, 10528), (10556, 10557),
# (10625, 10626), (10647, 10648), (10659, 10660), (10829, 10830),
# (10856, 10857), (10880, 10881), (10933, 10934), (10934, 10935),
# (11151, 11152), (11270, 11271), (11339, 11340), (11374, 11375),
# (11439, 11440), (11661, 11662), (11780, 11781), (11799, 11800),
# (11890, 11891), (11934, 11935), (11999, 12000), (12005, 12006),
# (12095, 12096), (12121, 12122), (12167, 12168), (12375, 12376),
# (12495, 12496), (12635, 12636), (12672, 12673), (13224, 13225),
# (13299, 13300), (13310, 13311), (13311, 13312), (13376, 13377),
# (13454, 13455), (13455, 13456), (13489, 13490), (13629, 13630),
# (13688, 13689), (13775, 13776), (13915, 13916), (13923, 13924),
# (14079, 14080), (14144, 14145), (14160, 14161), (14336, 14337),
# (14364, 14365), (14399, 14400), (14993, 14994), (15180, 15181),
# (15340, 15341), (15375, 15376), (15399, 15400), (15456, 15457),
# (15624, 15625), (15903, 15904), (15979, 15980), (16120, 16121),
# (16224, 16225), (16399, 16400), (16400, 16401), (16472, 16473),
# (16523, 16524), (16605, 16606), (16874, 16875), (16928, 16929),
# (17050, 17051), (17051, 17052), (17342, 17343), (17424, 17425),
# (17576, 17577), (17577, 17578), (17640, 17641), (17765, 17766),
# (17835, 17836), (17917, 17918), (18095, 18096), (18879, 18880),
# (18941, 18942), (19175, 19176), (19227, 19228), (19250, 19251),
# (19343, 19344), (19550, 19551), (19551, 19552), (19844, 19845),
# (19880, 19881), (19941, 19942), (19964, 19965), (20163, 20164),
# (20448, 20449), (20519, 20520), (20735, 20736), (20944, 20945),
# (21141, 21142), (21299, 21300), (21384, 21385), (21504, 21505),
# (21735, 21736), (22010, 22011), (22099, 22100), (22184, 22185),
# (22815, 22816), (22932, 22933), (23000, 23001), (23275, 23276),
# (23374, 23375), (23408, 23409), (23715, 23716), (23750, 23751),
# (24024, 24025), (24149, 24150), (24299, 24300), (24353, 24354),
# (24794, 24795), (25024, 25025), (25346, 25347), (25839, 25840),
# (25920, 25921), (26411, 26412), (26624, 26625), (26649, 26650),
# (26978, 26979), (26979, 26980), (26999, 27000), (27264, 27265),
# (27404, 27405), (27435, 27436), (27455, 27456), (28125, 28126),
# (28560, 28561), (28576, 28577), (28899, 28900), (29028, 29029),
# (29232, 29233), (29375, 29376), (29393, 29394), (29601, 29602),
# (29791, 29792), (30175, 30176), (30267, 30268), (30380, 30381),
# (30503, 30504), (30624, 30625), (30855, 30856), (30975, 30976),
# (31160, 31161), (31211, 31212), (31212, 31213), (31349, 31350),
# (31464, 31465), (31899, 31900), (32759, 32760), (32798, 32799),
# (32799, 32800), (33275, 33276), (33511, 33512), (34968, 34969),
# (35321, 35322), (35343, 35344), (35672, 35673), (35720, 35721),
# (36707, 36708), (36800, 36801), (36992, 36993), (37179, 37180),
# (37583, 37584), (37700, 37701), (38114, 38115), (39150, 39151),
# (40425, 40426), (40754, 40755), (40824, 40825), (41040, 41041),
# (41261, 41262), (41327, 41328), (41535, 41536), (41615, 41616),
# (41819, 41820), (41876, 41877), (42066, 42067), (42159, 42160),
# (42687, 42688), (43010, 43011), (43239, 43240), (43263, 43264),
# (43680, 43681), (43757, 43758), (44485, 44486), (44649, 44650),
# (45135, 45136), (45253, 45254), (45440, 45441), (45724, 45725),
# (46079, 46080), (46137, 46138), (46207, 46208), (46255, 46256),
# (46529, 46530), (46575, 46576), (47124, 47125), (47150, 47151),
# (48279, 48280), (48734, 48735), (48749, 48750), (49010, 49011),
# (49725, 49726), (49855, 49856), (49972, 49973), (50336, 50337),
# (50622, 50623), (52325, 52326), (52479, 52480), (53040, 53041),
# (53360, 53361), (53504, 53505), (54144, 54145), (55223, 55224),
# (55224, 55225), (55593, 55594), (56375, 56376), (56699, 56700),
# (57154, 57155), (58310, 58311), (60515, 60516), (61008, 61009),
# (61131, 61132), (61335, 61336), (62775, 62776), (63426, 63427),
# (64124, 64125), (66079, 66080), (66975, 66976), (67914, 67915),
# (70091, 70092), (70356, 70357), (70499, 70500), (70875, 70876),
# (71000, 71001), (71874, 71875), (71994, 71995), (72500, 72501),
# (72864, 72865), (73695, 73696), (75140, 75141), (75999, 76000),
# (76383, 76384), (76544, 76545), (76704, 76705), (76751, 76752),
# (76874, 76875), (77349, 77350), (78336, 78337), (78584, 78585),
# (81224, 81225), (81344, 81345), (81795, 81796), (82368, 82369),
# (82943, 82944), (83520, 83521), (83544, 83545), (85697, 85698),
# (87464, 87465), (89375, 89376), (90240, 90241), (90624, 90625),
# (91839, 91840), (91884, 91885), (93632, 93633), (94094, 94095),
# (94500, 94501), (95816, 95817), (95874, 95875), (96255, 96256),
# (96875, 96876), (97526, 97527), (98175, 98176), (98735, 98736),
# (99179, 99180), (99875, 99876), (100890, 100891),
# (101184, 101185), (102486, 102487), (102950, 102951),
# (103935, 103936), (104328, 104329), (104975, 104976),
# (105984, 105985), (106495, 106496), (107911, 107912),
# (108053, 108054), (108240, 108241), (108375, 108376),
# (109503, 109504), (110330, 110331), (111320, 111321),
# (111475, 111476), (111825, 111826), (112749, 112750),
# (116280, 116281), (116466, 116467), (116963, 116964),
# (117875, 117876), (118580, 118581), (119350, 119351),
# (119474, 119475), (121770, 121771), (121835, 121836),
# (122199, 122200), (122264, 122265), (122451, 122452),
# (122759, 122760), (123200, 123201), (124032, 124033),
# (124844, 124845), (126224, 126225), (129437, 129438),
# (130339, 130340), (130355, 130356), (130625, 130626),
# (133399, 133400), (134849, 134850), (136850, 136851),
# (138414, 138415), (140624, 140625), (141375, 141376),
# (142128, 142129), (142884, 142885), (147015, 147016),
# (151249, 151250), (152684, 152685), (152880, 152881),
# (153790, 153791), (156332, 156333), (158014, 158015),
# (158949, 158950), (163370, 163371), (164255, 164256),
# (165375, 165376), (166634, 166635), (166991, 166992),
# (167648, 167649), (168750, 168751), (169575, 169576),
# (171925, 171926), (174096, 174097), (175769, 175770),
# (176000, 176001), (176175, 176176), (176435, 176436),
# (178125, 178126), (180480, 180481), (180481, 180482),
# (181424, 181425), (184092, 184093), (190463, 190464),
# (191487, 191488), (193545, 193546), (194480, 194481),
# (194579, 194580), (196882, 196883), (201019, 201020),
# (202124, 202125), (203319, 203320), (203840, 203841),
# (205436, 205437), (207575, 207576), (210748, 210749),
# (210924, 210925), (212381, 212382), (212628, 212629),
# (214774, 214775), (216999, 217000), (227447, 227448),
# (228095, 228096), (228780, 228781), (230299, 230300),
# (230911, 230912), (233167, 233168), (235586, 235587),
# (240064, 240065), (240786, 240787), (241604, 241605),
# (242990, 242991), (243048, 243049), (244035, 244036),
# (245024, 245025), (246015, 246016), (248500, 248501),
# (254880, 254881), (260337, 260338), (264384, 264385),
# (265824, 265825), (268800, 268801), (275561, 275562),
# (275807, 275808), (278783, 278784), (282624, 282625),
# (283023, 283024), (283503, 283504), (284375, 284376),
# (288144, 288145), (290870, 290871), (301599, 301600),
# (302847, 302848), (303600, 303601), (304589, 304590),
# (305370, 305371), (306152, 306153), (312542, 312543),
# (314432, 314433), (328509, 328510), (330624, 330625),
# (331499, 331500), (336140, 336141), (344604, 344605),
# (345344, 345345), (346920, 346921), (347391, 347392),
# (369655, 369656), (369984, 369985), (374945, 374946),
# (386630, 386631), (395199, 395200), (400315, 400316),
# (404984, 404985), (407043, 407044), (408320, 408321),
# (410669, 410670), (412090, 412091), (414000, 414001),
# (416415, 416416), (418760, 418761), (421200, 421201),
# (422499, 422500), (424269, 424270), (452352, 452353),
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# ]
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