# List of all pairs of integers (x,y) such that
# x + y is a square, gcd(x,y) is square-free, x >= y, and the only primes dividing x and y lie in S,
# for all sets S = {2, 3, p},
# where p ranges over all other primes less or equal to 100.
# It contains 783 pairs (x,y), some of which may appear for more than one S.
# Format: "(x,y)".
# Computing this list took 1735 seconds.
# Authors: Rafael von Känel and Benjamin Matschke, 2015.
# License: Creative commons 3.0 by-nc.
#
#
# S = {2, 3, 5}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(5,-5)
(5,-4)
(5,-1)
(5,4)
(6,-6)
(6,-5)
(6,-2)
(6,3)
(8,1)
(9,-8)
(9,-5)
(10,-10)
(10,-9)
(10,-6)
(10,-1)
(10,6)
(12,-3)
(15,-15)
(15,-6)
(15,1)
(15,10)
(16,-15)
(16,9)
(18,-2)
(20,5)
(24,-15)
(24,1)
(25,-24)
(25,-16)
(25,-9)
(25,24)
(27,-2)
(30,-30)
(30,-5)
(30,6)
(40,-15)
(40,9)
(45,-20)
(45,4)
(48,1)
(50,-1)
(54,-50)
(54,-5)
(54,10)
(64,-15)
(75,6)
(80,1)
(81,-80)
(81,-32)
(81,40)
(90,10)
(96,-15)
(96,25)
(120,1)
(125,-4)
(144,25)
(150,-6)
(160,-135)
(160,9)
(225,64)
(240,-15)
(250,-81)
(250,-54)
(250,6)
(256,-135)
(288,1)
(360,1)
(384,-375)
(405,-5)
(486,-125)
(486,-2)
(625,-576)
(625,-96)
(625,216)
(640,-15)
(675,1)
(729,-200)
(729,640)
(960,1)
(1215,10)
(1536,-15)
(1600,81)
(2400,1)
(3456,25)
(4096,-375)
(4374,250)
(6250,-9)
(6561,-320)
(10240,-1215)
(15625,1536)
(25920,1)
(64000,9)
#
# S = {2, 3, 7}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(7,-7)
(7,-6)
(7,-3)
(7,2)
(8,-7)
(8,1)
(9,-8)
(9,7)
(12,-3)
(14,-14)
(14,2)
(16,-7)
(16,9)
(18,-14)
(18,-2)
(18,7)
(21,-21)
(21,-12)
(21,4)
(24,1)
(27,-2)
(28,-27)
(28,-3)
(28,21)
(32,-7)
(42,-42)
(42,-6)
(42,7)
(48,1)
(49,-48)
(49,-24)
(49,32)
(56,-7)
(63,-14)
(63,1)
(64,-63)
(72,49)
(81,-56)
(81,-32)
(84,-3)
(98,2)
(112,-63)
(112,9)
(128,-7)
(147,-3)
(162,-98)
(162,7)
(168,1)
(189,7)
(196,-27)
(224,1)
(288,1)
(343,-243)
(343,-54)
(343,18)
(448,-7)
(448,81)
(486,-2)
(512,-343)
(576,49)
(729,112)
(1024,-63)
(1372,-3)
(1458,-14)
(1792,-567)
(2401,-192)
(3024,1)
(3969,256)
(6561,-6272)
(9408,1)
(10368,2401)
(14406,-6)
(16128,1)
(16807,13122)
(30618,7)
(32768,-7)
#
# S = {2, 3, 11}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(11,-11)
(11,-2)
(12,-11)
(12,-3)
(16,9)
(18,-2)
(22,-22)
(22,-18)
(22,-6)
(22,3)
(24,1)
(27,-11)
(27,-2)
(27,22)
(33,-33)
(33,-32)
(33,-24)
(33,-8)
(33,3)
(33,16)
(36,-11)
(48,1)
(48,33)
(66,-66)
(66,-2)
(81,-32)
(88,33)
(88,81)
(99,1)
(99,22)
(121,-96)
(121,-72)
(121,48)
(132,-11)
(192,33)
(198,-2)
(243,-242)
(256,33)
(288,1)
(297,-176)
(297,-128)
(297,-8)
(297,64)
(352,9)
(363,-2)
(486,-2)
(528,1)
(726,3)
(729,-704)
(972,-11)
(1056,33)
(1089,-128)
(1728,121)
(2187,22)
(2662,-162)
(2673,-2048)
(2673,352)
(3993,-512)
(3993,-24)
(3993,768)
(4224,1)
(5632,297)
(12288,33)
(14641,243)
(19602,-2)
(24057,-32)
(59049,-968)
(235224,1)
(483153,-128)
#
# S = {2, 3, 13}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(13,-13)
(13,-12)
(13,-9)
(13,-4)
(13,3)
(13,12)
(16,9)
(18,-2)
(24,1)
(26,-26)
(26,-1)
(27,-26)
(27,-2)
(36,13)
(39,-39)
(39,-3)
(48,-39)
(48,1)
(52,-27)
(52,-3)
(64,-39)
(78,-78)
(78,3)
(81,-32)
(108,13)
(117,4)
(117,52)
(156,13)
(169,-144)
(169,-48)
(169,27)
(192,169)
(208,-39)
(208,81)
(243,13)
(288,1)
(486,-2)
(624,1)
(702,-26)
(729,-104)
(768,-39)
(832,9)
(2028,-3)
(2187,-338)
(2197,-972)
(2197,-81)
(2197,12)
(2808,1)
(3328,-3159)
(4212,13)
(16384,-3159)
(57122,-1)
(59049,13312)
(248832,169)
(851968,-39)
#
# S = {2, 3, 17}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(17,-17)
(17,-16)
(17,-8)
(17,-1)
(17,8)
(18,-17)
(18,-2)
(24,1)
(27,-2)
(32,17)
(34,-34)
(34,-18)
(34,-9)
(34,2)
(48,1)
(51,-51)
(51,-2)
(64,17)
(81,-32)
(81,-17)
(102,-102)
(102,-2)
(153,-128)
(153,-32)
(153,16)
(153,136)
(162,34)
(272,17)
(288,1)
(289,-288)
(289,-64)
(289,72)
(306,-17)
(486,-2)
(512,17)
(544,81)
(578,-2)
(1088,1)
(1224,1)
(1377,-1088)
(1377,-8)
(1377,1024)
(4913,128)
(6561,-4352)
(12393,-512)
(13122,-578)
(20736,289)
(34816,153)
(44217,-32768)
(70227,-2)
(83521,-1152)
(332928,1)
(1003833,2176)
#
# S = {2, 3, 19}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(19,-19)
(19,-18)
(19,-3)
(19,6)
(24,1)
(27,-2)
(38,-38)
(38,-2)
(48,1)
(54,-38)
(57,-57)
(57,-48)
(57,-32)
(57,-8)
(57,24)
(64,57)
(76,-27)
(81,-32)
(81,19)
(114,-114)
(171,-2)
(228,-3)
(288,1)
(304,57)
(342,19)
(361,-192)
(361,-72)
(384,57)
(486,-2)
(513,-512)
(513,-152)
(513,16)
(729,-608)
(864,361)
(1026,-2)
(1083,6)
(1216,9)
(1368,1)
(4617,-128)
(20577,-128)
(29184,57)
(39366,6859)
(41553,19456)
(65536,513)
(263169,-2048)
(1050624,1)
#
# S = {2, 3, 23}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(23,-23)
(23,2)
(24,-23)
(24,1)
(27,-23)
(27,-2)
(32,-23)
(46,-46)
(46,3)
(46,18)
(48,-23)
(48,1)
(54,46)
(69,-69)
(69,12)
(72,-23)
(81,-32)
(138,-138)
(138,6)
(144,-23)
(192,-23)
(243,46)
(256,-207)
(288,1)
(384,-23)
(486,-2)
(529,96)
(529,432)
(552,-23)
(621,-92)
(621,4)
(648,-23)
(729,-368)
(736,-207)
(864,-23)
(1152,529)
(1458,-1058)
(2048,-23)
(2208,1)
(2944,81)
(6912,-23)
(12288,-12167)
(13248,-23)
(17496,-12167)
(24334,2)
(34992,-23)
(89424,-23)
#
# S = {2, 3, 29}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(29,-29)
(29,-4)
(48,1)
(54,-29)
(58,-58)
(58,-54)
(58,-9)
(58,6)
(81,-32)
(87,-87)
(87,-6)
(96,-87)
(174,-174)
(256,-87)
(288,1)
(486,-2)
(729,232)
(783,1)
(783,58)
(841,-216)
(841,384)
(928,-87)
(1682,-1)
(3712,9)
(21141,-116)
(393216,-87)
#
# S = {2, 3, 31}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(31,-31)
(31,-27)
(31,-6)
(31,18)
(32,-31)
(48,1)
(62,-62)
(62,2)
(81,-32)
(93,-93)
(93,-12)
(124,-3)
(162,-62)
(186,-186)
(256,-31)
(288,1)
(486,-2)
(729,496)
(837,4)
(837,124)
(961,-432)
(961,128)
(992,-31)
(3968,1)
#
# S = {2, 3, 37}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(37,-37)
(37,-36)
(37,-12)
(37,-1)
(37,12)
(37,27)
(48,1)
(74,-74)
(81,-32)
(111,-111)
(148,-27)
(192,-111)
(222,-222)
(222,3)
(243,-74)
(288,1)
(324,37)
(444,-3)
(486,-2)
(1024,-999)
(1332,37)
(1369,-144)
(2368,-999)
(5328,1)
(6912,1369)
(9472,729)
(14348907,37)
#
# S = {2, 3, 41}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(41,-41)
(41,-32)
(41,-16)
(41,8)
(48,1)
(81,-32)
(82,-82)
(82,-81)
(82,-18)
(82,-1)
(82,18)
(123,-123)
(123,-2)
(128,41)
(162,-41)
(246,-246)
(288,1)
(369,-8)
(369,256)
(486,-2)
(1312,369)
(1681,-1152)
(1681,-81)
(3321,-512)
(3362,2)
(5043,-2)
(5248,81)
(6561,328)
(6642,82)
(26568,1)
(41984,41)
(7263027,-2)
(21789081,-8192)
#
# S = {2, 3, 43}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(43,-43)
(43,-27)
(43,-18)
(43,6)
(48,1)
(81,-32)
(86,-86)
(129,-129)
(129,-128)
(129,-48)
(129,-8)
(129,96)
(172,-3)
(258,-258)
(258,-2)
(288,1)
(486,-86)
(486,-2)
(486,43)
(1161,64)
(1161,688)
(1849,-1728)
(2752,729)
(3483,-2)
(4096,129)
(4374,-3698)
(16512,129)
(16641,-512)
(66048,1)
(94041,-8192)
#
# S = {2, 3, 47}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(47,-47)
(47,2)
(48,-47)
(48,1)
(72,-47)
(81,-32)
(94,-94)
(94,6)
(94,27)
(96,-47)
(128,-47)
(141,-141)
(141,3)
(162,94)
(216,-47)
(243,-47)
(282,-282)
(288,1)
(486,-2)
(576,-47)
(1728,-47)
(2209,192)
(2256,-47)
(3072,-47)
(4096,-3807)
(6016,-3807)
(9024,1)
(24064,6561)
(41472,2209)
(52488,-47)
(108288,-47)
(177147,94)
(5668704,-103823)
#
# S = {2, 3, 53}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(53,-53)
(53,-4)
(54,-53)
(81,-32)
(106,-106)
(106,-81)
(106,-6)
(159,-159)
(288,1)
(318,-318)
(318,6)
(384,-159)
(486,-2)
(729,-53)
(2809,216)
(2862,-53)
(11448,1)
(297754,-729)
#
# S = {2, 3, 59}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(59,-59)
(81,-32)
(108,-59)
(118,-118)
(118,-54)
(118,-18)
(118,3)
(177,-177)
(177,-128)
(177,-96)
(177,-8)
(177,48)
(243,118)
(288,1)
(354,-354)
(486,-2)
(531,-2)
(1593,256)
(1888,1593)
(3072,177)
(3481,-3456)
(7552,729)
(258066,-2)
(1161297,-131072)
#
# S = {2, 3, 61}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(61,-61)
(61,-36)
(61,-12)
(61,3)
(81,-32)
(108,61)
(122,-122)
(122,-1)
(183,-183)
(192,-183)
(243,-122)
(244,-243)
(288,1)
(366,-366)
(486,-2)
(732,-3)
(1024,-183)
(3721,768)
(3904,-183)
(14823,61)
(14884,-243)
(15616,9)
(59049,976)
(237168,1)
(1361886,3)
(7203978,-122)
#
# S = {2, 3, 67}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(67,-67)
(67,-18)
(67,-3)
(67,54)
(81,-32)
(134,-134)
(201,-201)
(201,-192)
(201,-32)
(201,24)
(268,-243)
(288,1)
(402,-402)
(402,-2)
(486,-2)
(1024,201)
(1809,-128)
(4288,201)
(4489,-768)
(17152,9)
(48843,-2)
(146529,-34304)
(73085409,-8)
#
# S = {2, 3, 71}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(71,-71)
(72,-71)
(81,-32)
(96,-71)
(142,-142)
(142,2)
(142,27)
(142,54)
(192,-71)
(213,-213)
(213,12)
(288,1)
(426,-426)
(432,-71)
(486,-2)
(512,-71)
(1296,-71)
(1458,142)
(2187,-71)
(5041,288)
(5112,-71)
(15123,6)
(20448,1)
(55296,-71)
#
# S = {2, 3, 73}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(73,-73)
(73,-72)
(73,-64)
(73,-48)
(73,-24)
(73,-9)
(73,8)
(73,27)
(73,48)
(81,-32)
(96,73)
(146,-146)
(146,-2)
(162,-146)
(216,73)
(219,-219)
(219,6)
(288,1)
(288,73)
(292,-243)
(292,-3)
(438,-438)
(438,3)
(486,-2)
(657,-128)
(657,-32)
(768,73)
(1024,657)
(1152,73)
(1296,73)
(4672,657)
(5256,73)
(5329,-2304)
(5329,-288)
(5329,2592)
(5913,-584)
(5913,16)
(6561,-2336)
(15552,73)
(18688,81)
(19683,-10658)
(21024,1)
(24576,73)
(53217,-32768)
(373248,73)
(389017,-34992)
(389017,294912)
(3501153,-512)
#
# S = {2, 3, 79}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(79,-79)
(79,-54)
(79,2)
(81,-32)
(128,-79)
(158,-158)
(162,-158)
(237,-237)
(237,-12)
(288,1)
(316,-27)
(474,-474)
(486,-2)
(6241,648)
(6399,-158)
(6399,1)
(6561,-632)
(1062882,79)
#
# S = {2, 3, 83}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(81,-32)
(83,-83)
(83,-2)
(108,-83)
(166,-166)
(166,-162)
(166,3)
(249,-249)
(249,-128)
(249,-24)
(249,192)
(288,1)
(486,-2)
(498,-498)
(2241,-32)
(6561,664)
(6723,1)
(6723,166)
(6889,-648)
(19683,-83)
(169984,2241)
#
# S = {2, 3, 89}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(81,-32)
(89,-89)
(89,-64)
(89,-8)
(89,32)
(178,-178)
(178,-162)
(178,-9)
(178,18)
(267,-267)
(288,1)
(486,-2)
(534,-534)
(801,-512)
(1458,-89)
(1602,-2)
(2403,-2)
(6561,2848)
(7209,16)
(7209,712)
(7921,-2592)
(8192,89)
(47526,-2)
#
# S = {2, 3, 97}:
#
(1,-1)
(2,-2)
(2,-1)
(2,2)
(3,-3)
(3,-2)
(3,1)
(4,-3)
(6,-6)
(6,-2)
(6,3)
(8,1)
(9,-8)
(12,-3)
(16,9)
(18,-2)
(24,1)
(27,-2)
(48,1)
(81,-32)
(97,-97)
(97,-96)
(97,-81)
(97,-72)
(97,-48)
(97,-16)
(97,3)
(97,24)
(97,72)
(128,97)
(192,97)
(194,-194)
(194,2)
(243,-194)
(288,1)
(291,-291)
(291,-2)
(388,-27)
(432,97)
(486,-2)
(582,-582)
(582,-6)
(864,97)
(873,-512)
(873,-32)
(2304,97)
(5832,97)
(6144,97)
(6561,6208)
(7857,64)
(7857,1552)
(9312,97)
(9409,-5184)
(9409,-384)
(28227,-3)
(37248,1)
(912673,-648)
(1589248,873)
(8957952,97)