## Puzzle – geometrical progression

To find 3 numbers,   $a, \; a \, r, \; a \, r^2$,   in geometrical progression such that each increased
by a given number   $n$   is a square.

math grad - Interest: Number theory
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### 2 Responses to Puzzle – geometrical progression

1. paul says:

Here is a list with a <= 500 and r <= 500^2

n^2 is ar^2 – ar and m^2 is ar – a

This list is not complete for any given {a}

a = 2, r = 9, n^2 = 144, m^2 = 16
a = 2, r = 289, n^2 = 166464, m^2 = 576
a = 2, r = 9801, n^2 = 192099600, m^2 = 19600
a = 3, r = 4, n^2 = 36, m^2 = 9
a = 3, r = 49, n^2 = 7056, m^2 = 144
a = 3, r = 676, n^2 = 1368900, m^2 = 2025
a = 3, r = 9409, n^2 = 265559616, m^2 = 28224
a = 3, r = 131044, n^2 = 51517196676, m^2 = 393129
a = 5, r = 81, n^2 = 32400, m^2 = 400
a = 5, r = 25921, n^2 = 3359361600, m^2 = 129600
a = 6, r = 25, n^2 = 3600, m^2 = 144
a = 6, r = 2401, n^2 = 34574400, m^2 = 14400
a = 6, r = 235225, n^2 = 331983392400, m^2 = 1411344
a = 7, r = 64, n^2 = 28224, m^2 = 441
a = 7, r = 16129, n^2 = 1820899584, m^2 = 112896
a = 8, r = 9, n^2 = 576, m^2 = 64
a = 8, r = 289, n^2 = 665856, m^2 = 2304
a = 8, r = 9801, n^2 = 768398400, m^2 = 78400
a = 10, r = 361, n^2 = 1299600, m^2 = 3600
a = 11, r = 100, n^2 = 108900, m^2 = 1089
a = 11, r = 39601, n^2 = 17250195600, m^2 = 435600
a = 12, r = 4, n^2 = 144, m^2 = 36
a = 12, r = 49, n^2 = 28224, m^2 = 576
a = 12, r = 676, n^2 = 5475600, m^2 = 8100
a = 12, r = 9409, n^2 = 1062238464, m^2 = 112896
a = 12, r = 131044, n^2 = 206068786704, m^2 = 1572516
a = 14, r = 225, n^2 = 705600, m^2 = 3136
a = 14, r = 201601, n^2 = 568998662400, m^2 = 2822400
a = 15, r = 16, n^2 = 3600, m^2 = 225
a = 15, r = 961, n^2 = 13838400, m^2 = 14400
a = 15, r = 59536, n^2 = 53167136400, m^2 = 893025
a = 17, r = 1089, n^2 = 20142144, m^2 = 18496
a = 18, r = 9, n^2 = 1296, m^2 = 144
a = 18, r = 289, n^2 = 1498176, m^2 = 5184
a = 18, r = 9801, n^2 = 1728896400, m^2 = 176400
a = 19, r = 28900, n^2 = 15868440900, m^2 = 549081
a = 20, r = 81, n^2 = 129600, m^2 = 1600
a = 20, r = 25921, n^2 = 13437446400, m^2 = 518400
a = 21, r = 3025, n^2 = 192099600, m^2 = 63504
a = 22, r = 38809, n^2 = 33134192784, m^2 = 853776
a = 23, r = 576, n^2 = 7617600, m^2 = 13225
a = 24, r = 25, n^2 = 14400, m^2 = 576
a = 24, r = 2401, n^2 = 138297600, m^2 = 57600
a = 24, r = 235225, n^2 = 1327933569600, m^2 = 5645376
a = 26, r = 2601, n^2 = 175827600, m^2 = 67600
a = 27, r = 4, n^2 = 324, m^2 = 81
a = 27, r = 49, n^2 = 63504, m^2 = 1296
a = 27, r = 676, n^2 = 12320100, m^2 = 18225
a = 27, r = 9409, n^2 = 2390036544, m^2 = 254016
a = 27, r = 131044, n^2 = 463654770084, m^2 = 3538161
a = 28, r = 64, n^2 = 112896, m^2 = 1764
a = 28, r = 16129, n^2 = 7283598336, m^2 = 451584
a = 30, r = 121, n^2 = 435600, m^2 = 3600
a = 30, r = 58081, n^2 = 101200334400, m^2 = 1742400
a = 32, r = 9, n^2 = 2304, m^2 = 256
a = 32, r = 289, n^2 = 2663424, m^2 = 9216
a = 32, r = 9801, n^2 = 3073593600, m^2 = 313600
a = 33, r = 529, n^2 = 9217296, m^2 = 17424
a = 34, r = 1225, n^2 = 50979600, m^2 = 41616
a = 35, r = 36, n^2 = 44100, m^2 = 1225
a = 35, r = 5041, n^2 = 889232400, m^2 = 176400
a = 37, r = 5329, n^2 = 1050537744, m^2 = 197136
a = 38, r = 1369, n^2 = 71166096, m^2 = 51984
a = 39, r = 625, n^2 = 15210000, m^2 = 24336
a = 40, r = 361, n^2 = 5198400, m^2 = 14400
a = 42, r = 169, n^2 = 1192464, m^2 = 7056
a = 42, r = 113569, n^2 = 541707776064, m^2 = 4769856
a = 44, r = 100, n^2 = 435600, m^2 = 4356
a = 44, r = 39601, n^2 = 69000782400, m^2 = 1742400
a = 45, r = 81, n^2 = 291600, m^2 = 3600
a = 45, r = 25921, n^2 = 30234254400, m^2 = 1166400
a = 47, r = 2304, n^2 = 249387264, m^2 = 108241
a = 48, r = 4, n^2 = 576, m^2 = 144
a = 48, r = 49, n^2 = 112896, m^2 = 2304
a = 48, r = 676, n^2 = 21902400, m^2 = 32400
a = 48, r = 9409, n^2 = 4248953856, m^2 = 451584
a = 48, r = 131044, n^2 = 824275146816, m^2 = 6290064
a = 50, r = 9, n^2 = 3600, m^2 = 400
a = 50, r = 289, n^2 = 4161600, m^2 = 14400
a = 50, r = 9801, n^2 = 4802490000, m^2 = 490000
a = 51, r = 2500, n^2 = 318622500, m^2 = 127449
a = 54, r = 25, n^2 = 32400, m^2 = 1296
a = 54, r = 2401, n^2 = 311169600, m^2 = 129600
a = 54, r = 235225, n^2 = 2987850531600, m^2 = 12702096
a = 55, r = 7921, n^2 = 3450387600, m^2 = 435600
a = 56, r = 225, n^2 = 2822400, m^2 = 12544
a = 56, r = 201601, n^2 = 2275994649600, m^2 = 11289600
a = 57, r = 22801, n^2 = 29632179600, m^2 = 1299600
a = 60, r = 16, n^2 = 14400, m^2 = 900
a = 60, r = 961, n^2 = 55353600, m^2 = 57600
a = 60, r = 59536, n^2 = 212668545600, m^2 = 3572100
a = 62, r = 3969, n^2 = 976437504, m^2 = 246016
a = 63, r = 64, n^2 = 254016, m^2 = 3969
a = 63, r = 16129, n^2 = 16388096256, m^2 = 1016064
a = 65, r = 16641, n^2 = 17998905600, m^2 = 1081600
a = 66, r = 4225, n^2 = 1177862400, m^2 = 278784
a = 68, r = 1089, n^2 = 80568576, m^2 = 73984
a = 70, r = 63001, n^2 = 277834410000, m^2 = 4410000
a = 72, r = 9, n^2 = 5184, m^2 = 576
a = 72, r = 289, n^2 = 5992704, m^2 = 20736
a = 72, r = 9801, n^2 = 6915585600, m^2 = 705600
a = 75, r = 4, n^2 = 900, m^2 = 225
a = 75, r = 49, n^2 = 176400, m^2 = 3600
a = 75, r = 676, n^2 = 34222500, m^2 = 50625
a = 75, r = 9409, n^2 = 6638990400, m^2 = 705600
a = 75, r = 131044, n^2 = 1287929916900, m^2 = 9828225
a = 76, r = 28900, n^2 = 63473763600, m^2 = 2196324
a = 77, r = 123201, n^2 = 1168733966400, m^2 = 9486400
a = 78, r = 2809, n^2 = 615238416, m^2 = 219024
a = 79, r = 6400, n^2 = 3235334400, m^2 = 505521
a = 80, r = 81, n^2 = 518400, m^2 = 6400
a = 80, r = 25921, n^2 = 53749785600, m^2 = 2073600
a = 82, r = 26569, n^2 = 57882585744, m^2 = 2178576
a = 83, r = 6724, n^2 = 3752052516, m^2 = 558009
a = 84, r = 3025, n^2 = 768398400, m^2 = 254016
a = 87, r = 784, n^2 = 53406864, m^2 = 68121
a = 88, r = 38809, n^2 = 132536771136, m^2 = 3415104
a = 90, r = 361, n^2 = 11696400, m^2 = 32400
a = 92, r = 576, n^2 = 30470400, m^2 = 52900
a = 95, r = 1521, n^2 = 219632400, m^2 = 144400
a = 96, r = 25, n^2 = 57600, m^2 = 2304
a = 96, r = 2401, n^2 = 553190400, m^2 = 230400
a = 96, r = 235225, n^2 = 5311734278400, m^2 = 22581504
a = 98, r = 9, n^2 = 7056, m^2 = 784
a = 98, r = 289, n^2 = 8156736, m^2 = 28224
a = 98, r = 9801, n^2 = 9412880400, m^2 = 960400
a = 99, r = 100, n^2 = 980100, m^2 = 9801
a = 99, r = 39601, n^2 = 155251760400, m^2 = 3920400
a = 101, r = 40401, n^2 = 164852240400, m^2 = 4080400
a = 102, r = 10201, n^2 = 10613120400, m^2 = 1040400
a = 104, r = 2601, n^2 = 703310400, m^2 = 270400
a = 105, r = 1681, n^2 = 296528400, m^2 = 176400
a = 108, r = 4, n^2 = 1296, m^2 = 324
a = 108, r = 49, n^2 = 254016, m^2 = 5184
a = 108, r = 676, n^2 = 49280400, m^2 = 72900
a = 108, r = 9409, n^2 = 9560146176, m^2 = 1016064
a = 108, r = 131044, n^2 = 1854619080336, m^2 = 14152644
a = 110, r = 441, n^2 = 21344400, m^2 = 48400
a = 111, r = 87025, n^2 = 840632259600, m^2 = 9659664
a = 112, r = 64, n^2 = 451584, m^2 = 7056
a = 112, r = 16129, n^2 = 29134393344, m^2 = 1806336
a = 119, r = 14400, n^2 = 24674126400, m^2 = 1713481
a = 120, r = 121, n^2 = 1742400, m^2 = 14400
a = 120, r = 58081, n^2 = 404801337600, m^2 = 6969600
a = 122, r = 59049, n^2 = 425380492944, m^2 = 7203856
a = 123, r = 14884, n^2 = 27246784356, m^2 = 1830609
a = 125, r = 81, n^2 = 810000, m^2 = 10000
a = 125, r = 25921, n^2 = 83984040000, m^2 = 3240000
a = 126, r = 225, n^2 = 6350400, m^2 = 28224
a = 126, r = 201601, n^2 = 5120987961600, m^2 = 25401600
a = 128, r = 9, n^2 = 9216, m^2 = 1024
a = 128, r = 289, n^2 = 10653696, m^2 = 36864
a = 128, r = 9801, n^2 = 12294374400, m^2 = 1254400
a = 132, r = 529, n^2 = 36869184, m^2 = 69696
a = 135, r = 16, n^2 = 32400, m^2 = 2025
a = 135, r = 961, n^2 = 124545600, m^2 = 129600
a = 135, r = 59536, n^2 = 478504227600, m^2 = 8037225
a = 136, r = 1225, n^2 = 203918400, m^2 = 166464
a = 138, r = 2209, n^2 = 673091136, m^2 = 304704
a = 140, r = 36, n^2 = 176400, m^2 = 4900
a = 140, r = 5041, n^2 = 3556929600, m^2 = 705600
a = 141, r = 9025, n^2 = 11483265600, m^2 = 1272384
a = 142, r = 20449, n^2 = 59376043584, m^2 = 2903616
a = 143, r = 144, n^2 = 2944656, m^2 = 20449
a = 143, r = 82369, n^2 = 970193480256, m^2 = 11778624
a = 145, r = 83521, n^2 = 1011472718400, m^2 = 12110400
a = 146, r = 21025, n^2 = 64536321600, m^2 = 3069504
a = 147, r = 4, n^2 = 1764, m^2 = 441
a = 147, r = 49, n^2 = 345744, m^2 = 7056
a = 147, r = 676, n^2 = 67076100, m^2 = 99225
a = 147, r = 9409, n^2 = 13012421184, m^2 = 1382976
a = 147, r = 131044, n^2 = 2524342637124, m^2 = 19263321
a = 148, r = 5329, n^2 = 4202150976, m^2 = 788544
a = 150, r = 25, n^2 = 90000, m^2 = 3600
a = 150, r = 2401, n^2 = 864360000, m^2 = 360000
a = 150, r = 235225, n^2 = 8299584810000, m^2 = 35283600
a = 152, r = 1369, n^2 = 284664384, m^2 = 207936
a = 153, r = 1089, n^2 = 181279296, m^2 = 166464
a = 155, r = 62001, n^2 = 595829610000, m^2 = 9610000
a = 156, r = 625, n^2 = 60840000, m^2 = 97344
a = 160, r = 361, n^2 = 20793600, m^2 = 57600
a = 162, r = 9, n^2 = 11664, m^2 = 1296
a = 162, r = 289, n^2 = 13483584, m^2 = 46656
a = 162, r = 9801, n^2 = 15560067600, m^2 = 1587600
a = 167, r = 28224, n^2 = 133026513984, m^2 = 4713241
a = 168, r = 169, n^2 = 4769856, m^2 = 28224
a = 168, r = 113569, n^2 = 2166831104256, m^2 = 19079424
a = 170, r = 114921, n^2 = 2245142624400, m^2 = 19536400
a = 171, r = 28900, n^2 = 142815968100, m^2 = 4941729
a = 175, r = 64, n^2 = 705600, m^2 = 11025
a = 175, r = 16129, n^2 = 45522489600, m^2 = 2822400
a = 176, r = 100, n^2 = 1742400, m^2 = 17424
a = 176, r = 39601, n^2 = 276003129600, m^2 = 6969600
a = 180, r = 81, n^2 = 1166400, m^2 = 14400
a = 180, r = 25921, n^2 = 120937017600, m^2 = 4665600
a = 182, r = 729, n^2 = 96589584, m^2 = 132496
a = 183, r = 237169, n^2 = 10293548222736, m^2 = 43401744
a = 188, r = 2304, n^2 = 997549056, m^2 = 432964
a = 189, r = 3025, n^2 = 1728896400, m^2 = 571536
a = 192, r = 4, n^2 = 2304, m^2 = 576
a = 192, r = 49, n^2 = 451584, m^2 = 9216
a = 192, r = 676, n^2 = 87609600, m^2 = 129600
a = 192, r = 9409, n^2 = 16995815424, m^2 = 1806336
a = 192, r = 131044, n^2 = 3297100587264, m^2 = 25160256
a = 194, r = 38025, n^2 = 280497344400, m^2 = 7376656
a = 195, r = 196, n^2 = 7452900, m^2 = 38025
a = 195, r = 152881, n^2 = 4557627219600, m^2 = 29811600
a = 197, r = 154449, n^2 = 4699304812944, m^2 = 30426256
a = 198, r = 38809, n^2 = 298207735056, m^2 = 7683984
a = 200, r = 9, n^2 = 14400, m^2 = 1600
a = 200, r = 289, n^2 = 16646400, m^2 = 57600
a = 200, r = 9801, n^2 = 19209960000, m^2 = 1960000
a = 203, r = 3249, n^2 = 2142208656, m^2 = 659344
a = 204, r = 2500, n^2 = 1274490000, m^2 = 509796
a = 207, r = 576, n^2 = 68558400, m^2 = 119025
a = 210, r = 841, n^2 = 148352400, m^2 = 176400
a = 215, r = 1936, n^2 = 805424400, m^2 = 416025
a = 216, r = 25, n^2 = 129600, m^2 = 5184
a = 216, r = 2401, n^2 = 1244678400, m^2 = 518400
a = 216, r = 235225, n^2 = 11951402126400, m^2 = 50808384
a = 219, r = 5476, n^2 = 6565860900, m^2 = 1199025
a = 220, r = 7921, n^2 = 13801550400, m^2 = 1742400
a = 222, r = 22201, n^2 = 109415408400, m^2 = 4928400
a = 223, r = 50176, n^2 = 561420518400, m^2 = 11189025
a = 224, r = 225, n^2 = 11289600, m^2 = 50176
a = 224, r = 201601, n^2 = 9103978598400, m^2 = 45158400
a = 226, r = 203401, n^2 = 9350018528400, m^2 = 45968400
a = 227, r = 51076, n^2 = 592176420900, m^2 = 11594025
a = 228, r = 22801, n^2 = 118528718400, m^2 = 5198400
a = 230, r = 8281, n^2 = 15770336400, m^2 = 1904400
a = 231, r = 5776, n^2 = 7705328400, m^2 = 1334025
a = 234, r = 2601, n^2 = 1582448400, m^2 = 608400
a = 235, r = 2116, n^2 = 1051704900, m^2 = 497025
a = 240, r = 16, n^2 = 57600, m^2 = 3600
a = 240, r = 961, n^2 = 221414400, m^2 = 230400
a = 240, r = 59536, n^2 = 850674182400, m^2 = 14288400
a = 242, r = 9, n^2 = 17424, m^2 = 1936
a = 242, r = 289, n^2 = 20142144, m^2 = 69696
a = 242, r = 9801, n^2 = 23244051600, m^2 = 2371600
a = 243, r = 4, n^2 = 2916, m^2 = 729
a = 243, r = 49, n^2 = 571536, m^2 = 11664
a = 243, r = 676, n^2 = 110880900, m^2 = 164025
a = 243, r = 9409, n^2 = 21510328896, m^2 = 2286144
a = 243, r = 131044, n^2 = 4172892930756, m^2 = 31843449
a = 245, r = 81, n^2 = 1587600, m^2 = 19600
a = 245, r = 25921, n^2 = 164608718400, m^2 = 6350400
a = 248, r = 3969, n^2 = 3905750016, m^2 = 984064
a = 250, r = 361, n^2 = 32490000, m^2 = 90000
a = 252, r = 64, n^2 = 1016064, m^2 = 15876
a = 252, r = 16129, n^2 = 65552385024, m^2 = 4064256
a = 254, r = 65025, n^2 = 1073959142400, m^2 = 16516096
a = 255, r = 256, n^2 = 16646400, m^2 = 65025
a = 258, r = 66049, n^2 = 1125500322816, m^2 = 17040384
a = 260, r = 16641, n^2 = 71995622400, m^2 = 4326400
a = 264, r = 4225, n^2 = 4711449600, m^2 = 1115136
a = 270, r = 121, n^2 = 3920400, m^2 = 32400
a = 270, r = 58081, n^2 = 910803009600, m^2 = 15681600
a = 272, r = 1089, n^2 = 322274304, m^2 = 295936
a = 275, r = 100, n^2 = 2722500, m^2 = 27225
a = 275, r = 39601, n^2 = 431254890000, m^2 = 10890000
a = 280, r = 63001, n^2 = 1111337640000, m^2 = 17640000
a = 287, r = 82944, n^2 = 1974452143104, m^2 = 23804641
a = 288, r = 9, n^2 = 20736, m^2 = 2304
a = 288, r = 289, n^2 = 23970816, m^2 = 82944
a = 288, r = 9801, n^2 = 27662342400, m^2 = 2822400
a = 291, r = 84100, n^2 = 2058163236900, m^2 = 24472809
a = 294, r = 25, n^2 = 176400, m^2 = 7056
a = 294, r = 2401, n^2 = 1694145600, m^2 = 705600
a = 294, r = 235225, n^2 = 16267186227600, m^2 = 69155856
a = 297, r = 529, n^2 = 82955664, m^2 = 156816
a = 299, r = 172225, n^2 = 8868722241600, m^2 = 51494976
a = 300, r = 4, n^2 = 3600, m^2 = 900
a = 300, r = 49, n^2 = 705600, m^2 = 14400
a = 300, r = 676, n^2 = 136890000, m^2 = 202500
a = 300, r = 9409, n^2 = 26555961600, m^2 = 2822400
a = 300, r = 131044, n^2 = 5151719667600, m^2 = 39312900
a = 304, r = 28900, n^2 = 253895054400, m^2 = 8785296
a = 305, r = 239121, n^2 = 17439477123600, m^2 = 72931600
a = 306, r = 1225, n^2 = 458816400, m^2 = 374544
a = 308, r = 123201, n^2 = 4674935865600, m^2 = 37945600
a = 312, r = 2809, n^2 = 2460953664, m^2 = 876096
a = 315, r = 36, n^2 = 396900, m^2 = 11025
a = 315, r = 5041, n^2 = 8003091600, m^2 = 1587600
a = 316, r = 6400, n^2 = 12941337600, m^2 = 2022084
a = 318, r = 11449, n^2 = 41679672336, m^2 = 3640464
a = 320, r = 81, n^2 = 2073600, m^2 = 25600
a = 320, r = 25921, n^2 = 214999142400, m^2 = 8294400
a = 321, r = 46225, n^2 = 685882112400, m^2 = 14837904
a = 322, r = 104329, n^2 = 3504788363664, m^2 = 33593616
a = 323, r = 324, n^2 = 33802596, m^2 = 104329
a = 326, r = 105625, n^2 = 3637030410000, m^2 = 34433424
a = 327, r = 47089, n^2 = 725065874064, m^2 = 15397776
a = 328, r = 26569, n^2 = 231530342976, m^2 = 8714304
a = 330, r = 11881, n^2 = 46578272400, m^2 = 3920400
a = 332, r = 6724, n^2 = 15008210064, m^2 = 2232036
a = 333, r = 5329, n^2 = 9454839696, m^2 = 1774224
a = 336, r = 3025, n^2 = 3073593600, m^2 = 1016064
a = 338, r = 9, n^2 = 24336, m^2 = 2704
a = 338, r = 289, n^2 = 28132416, m^2 = 97344
a = 338, r = 9801, n^2 = 32464832400, m^2 = 3312400
a = 342, r = 1369, n^2 = 640494864, m^2 = 467856
a = 343, r = 64, n^2 = 1382976, m^2 = 21609
a = 343, r = 16129, n^2 = 89224079616, m^2 = 5531904
a = 348, r = 784, n^2 = 213627456, m^2 = 272484
a = 350, r = 225, n^2 = 17640000, m^2 = 78400
a = 350, r = 201601, n^2 = 14224966560000, m^2 = 70560000
a = 351, r = 625, n^2 = 136890000, m^2 = 219024
a = 352, r = 38809, n^2 = 530147084544, m^2 = 13660416
a = 359, r = 129600, n^2 = 6029774913600, m^2 = 46526041
a = 360, r = 361, n^2 = 46785600, m^2 = 129600
a = 363, r = 4, n^2 = 4356, m^2 = 1089
a = 363, r = 49, n^2 = 853776, m^2 = 17424
a = 363, r = 676, n^2 = 165636900, m^2 = 245025
a = 363, r = 9409, n^2 = 32132713536, m^2 = 3415104
a = 363, r = 131044, n^2 = 6233580797796, m^2 = 47568609
a = 368, r = 576, n^2 = 121881600, m^2 = 211600
a = 375, r = 16, n^2 = 90000, m^2 = 5625
a = 375, r = 961, n^2 = 345960000, m^2 = 360000
a = 375, r = 59536, n^2 = 1329178410000, m^2 = 22325625
a = 377, r = 54289, n^2 = 1111109944464, m^2 = 20466576
a = 378, r = 169, n^2 = 10732176, m^2 = 63504
a = 378, r = 113569, n^2 = 4875369984576, m^2 = 42928704
a = 380, r = 1521, n^2 = 878529600, m^2 = 577600
a = 384, r = 25, n^2 = 230400, m^2 = 9216
a = 384, r = 2401, n^2 = 2212761600, m^2 = 921600
a = 384, r = 235225, n^2 = 21246937113600, m^2 = 90326016
a = 390, r = 6241, n^2 = 15188097600, m^2 = 2433600
a = 392, r = 9, n^2 = 28224, m^2 = 3136
a = 392, r = 289, n^2 = 32626944, m^2 = 112896
a = 392, r = 9801, n^2 = 37651521600, m^2 = 3841600
a = 395, r = 25281, n^2 = 252445953600, m^2 = 9985600
a = 396, r = 100, n^2 = 3920400, m^2 = 39204
a = 396, r = 39601, n^2 = 621007041600, m^2 = 15681600
a = 398, r = 159201, n^2 = 10087230081600, m^2 = 63361600
a = 399, r = 400, n^2 = 63680400, m^2 = 159201
a = 402, r = 160801, n^2 = 10394433921600, m^2 = 64641600
a = 404, r = 40401, n^2 = 659408961600, m^2 = 16321600
a = 405, r = 81, n^2 = 2624400, m^2 = 32400
a = 405, r = 25921, n^2 = 272108289600, m^2 = 10497600
a = 408, r = 10201, n^2 = 42452481600, m^2 = 4161600
a = 410, r = 6561, n^2 = 17646465600, m^2 = 2689600
a = 416, r = 2601, n^2 = 2813241600, m^2 = 1081600
a = 420, r = 1681, n^2 = 1186113600, m^2 = 705600
a = 423, r = 2304, n^2 = 2244485376, m^2 = 974169
a = 425, r = 1089, n^2 = 503553600, m^2 = 462400
a = 427, r = 3844, n^2 = 6307854084, m^2 = 1640961
a = 432, r = 4, n^2 = 5184, m^2 = 1296
a = 432, r = 49, n^2 = 1016064, m^2 = 20736
a = 432, r = 676, n^2 = 197121600, m^2 = 291600
a = 432, r = 9409, n^2 = 38240584704, m^2 = 4064256
a = 432, r = 131044, n^2 = 7418476321344, m^2 = 56610576
a = 434, r = 15625, n^2 = 105950250000, m^2 = 6780816
a = 435, r = 21316, n^2 = 197642484900, m^2 = 9272025
a = 438, r = 85849, n^2 = 3228044648976, m^2 = 37601424
a = 439, r = 193600, n^2 = 16454056449600, m^2 = 84989961
a = 440, r = 441, n^2 = 85377600, m^2 = 193600
a = 443, r = 195364, n^2 = 16907935429476, m^2 = 86545809
a = 444, r = 87025, n^2 = 3362529038400, m^2 = 38638656
a = 447, r = 21904, n^2 = 214454200464, m^2 = 9790641
a = 448, r = 64, n^2 = 1806336, m^2 = 28224
a = 448, r = 16129, n^2 = 116537573376, m^2 = 7225344
a = 450, r = 9, n^2 = 32400, m^2 = 3600
a = 450, r = 289, n^2 = 37454400, m^2 = 129600
a = 450, r = 9801, n^2 = 43222410000, m^2 = 4410000
a = 455, r = 4096, n^2 = 7631769600, m^2 = 1863225
a = 459, r = 2500, n^2 = 2867602500, m^2 = 1147041
a = 462, r = 1849, n^2 = 1578631824, m^2 = 853776
a = 473, r = 7569, n^2 = 27094476816, m^2 = 3579664
a = 475, r = 28900, n^2 = 396711022500, m^2 = 13727025
a = 476, r = 14400, n^2 = 98696505600, m^2 = 6853924
a = 480, r = 121, n^2 = 6969600, m^2 = 57600
a = 480, r = 58081, n^2 = 1619205350400, m^2 = 27878400
a = 482, r = 233289, n^2 = 26232138679824, m^2 = 112444816
a = 483, r = 484, n^2 = 112911876, m^2 = 233289
a = 486, r = 25, n^2 = 291600, m^2 = 11664
a = 486, r = 2401, n^2 = 2800526400, m^2 = 1166400
a = 486, r = 235225, n^2 = 26890654784400, m^2 = 114318864
a = 488, r = 59049, n^2 = 1701521971776, m^2 = 28815424
a = 490, r = 361, n^2 = 63680400, m^2 = 176400
a = 492, r = 14884, n^2 = 108987137424, m^2 = 7322436
a = 495, r = 7921, n^2 = 31053488400, m^2 = 3920400
a = 500, r = 81, n^2 = 3240000, m^2 = 40000
a = 500, r = 25921, n^2 = 335936160000, m^2 = 12960000

Paul.

• benvitalis says:

You need to find n such that a+n, a*r + n, a*r^2 + n are all squares