To make {(a+b-c), (a+c-b), (b+c-a)} all squares, (a,b,c) in AP

 
 

Find three numbers   a, \; b, \; c   in arithmetic progression such that

(1)    a \; + \; b \; - \; c \; = \; x^2
(2)    a \; + \; c \; - \; b \; = \; y^2
(3)    b \; + \; c \; - \; a \; = \; z^2

 
a,     b = a + k,     c = a + 2 \, k
where the common difference is   k

(1)    a + (a + k) - (a + 2 \, k) = x^2   ………..   a - k = x^2
(2)    a + (a + 2 \, k) - (a + k) = y^2   ………..   a + k = y^2
(3)    (a + k) + (a + 2 \, k) - a = z^2   ………..   a + 3 \, k = z^2

x^2,   y^2,   z^2   are in A.P.,   that is,     2 \, y^2 = x^2 + z^2

the common difference is   2 \, k
 

 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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4 Responses to To make {(a+b-c), (a+c-b), (b+c-a)} all squares, (a,b,c) in AP

  1. paul says:

    Here are a few, Format is {a, b, c}

    {52,100,148}
    {109,169,229}
    {117,225,333}
    {169,289,409}
    {208,400,592}
    {325,625,925}
    {421,841,1261}
    {436,676,916}
    {457,625,793}
    {468,900,1332}
    {637,1225,1813}
    {676,1156,1636}
    {832,1600,2368}
    {949,1369,1789}
    {981,1521,2061}
    {1053,2025,2997}
    {1300,2500,3700}
    {1321,1681,2041}
    {1521,2601,3681}
    {1549,2809,4069}
    {1573,3025,4477}
    {1684,3364,5044}
    {1744,2704,3664}
    {1828,2500,3172}
    {1872,3600,5328}
    {2197,4225,6253}
    {2377,4225,6073}
    {2548,4900,7252}
    {2689,5329,7969}
    {2704,4624,6544}
    {2725,4225,5725}
    {2925,5625,8325}
    {3061,3721,4381}
    {3217,4225,5233}
    {3328,6400,9472}
    {3757,7225,10693}
    {3789,7569,11349}
    {3796,5476,7156}
    {3924,6084,8244}
    {4113,5625,7137}
    {4212,8100,11988}
    {4225,7225,10225}
    {4453,7225,9997}
    {4693,9025,13357}
    {4729,9409,14089}
    {4801,7921,11041}
    {5200,10000,14800}
    {5284,6724,8164}
    {5341,8281,11221}
    {5733,11025,16317}
    {6084,10404,14724}
    {6133,7225,8317}
    {6196,11236,16276}
    {6292,12100,17908}
    {6421,11881,17341}
    {6736,13456,20176}
    {6877,13225,19573}
    {6976,10816,14656}
    {7312,10000,12688}
    {7488,14400,21312}
    {8125,15625,23125}
    {8221,10201,12181}
    {8281,14161,20041}
    {8541,12321,16101}
    {8788,16900,25012}
    {8829,13689,18549}
    {9477,18225,26973}
    {9508,16900,24292}
    {9529,18769,28009}
    {10192,19600,29008}
    {10477,15625,20773}
    {10525,21025,31525}
    {10756,21316,31876}
    {10816,18496,26176}
    {10900,16900,22900}
    {10933,21025,31117}
    {11089,12769,14449}
    {11425,15625,19825}
    {11700,22500,33300}
    {11889,15129,18369}
    {12244,14884,17524}
    {12493,24025,35557}
    {12868,16900,20932}
    {13189,20449,27709}
    {13312,25600,37888}
    {13429,24649,35869}
    {13689,23409,33129}
    {13941,25281,36621}
    {14157,27225,40293}
    {15061,22201,29341}
    {15184,21904,28624}
    {15696,24336,32976}
    {16452,22500,28548}
    {16900,28900,40900}
    {17593,21025,24457}
    {17812,28900,39988}
    {18421,28561,38701}
    {18577,21025,23473}
    {19204,31684,44164}
    {20449,34969,49489}
    {21136,26896,32656}
    {21289,37249,53209}
    {21349,29929,38509}
    {21364,33124,44884}
    {21393,38025,54657}
    {22393,30625,38857}
    {23725,34225,44725}
    {24193,34225,44257}
    {24336,41616,58896}
    {24525,38025,51525}
    {24532,28900,33268}
    {26317,42025,57733}
    {27549,33489,39429}
    {27904,43264,58624}
    {28953,38025,47097}
    {29248,40000,50752}
    {29341,32761,36181}
    {31501,48841,66181}
    {32884,40804,48724}
    {33025,42025,51025}
    {33349,38809,44269}
    {34164,49284,64404}
    {35316,54756,74196}
    {37017,50625,64233}
    {39181,52441,65701}
    {39349,61009,82669}
    {41908,62500,83092}
    {43600,67600,91600}
    {44221,48841,53461}
    {44356,51076,57796}
    {45700,62500,79300}
    {46501,67081,87661}
    {46513,70225,93937}
    {47556,60516,73476}
    {48976,59536,70096}
    {51472,67600,83728}
    {54421,72361,90301}
    {55197,65025,74853}
    {55297,75625,95953}
    {57889,66049,74209}
    {60736,87616,114496}
    {64153,70225,76297}
    {64729,82369,100009}
    {65808,90000,114192}
    {66469,85849,105229}
    {70372,84100,97828}
    {73989,91809,109629}
    {74308,84100,93892}
    {76525,93025,109525}
    {77233,105625,134017}
    {77389,100489,123589}
    {80425,105625,130825}
    {84544,107584,130624}
    {89572,122500,155428}
    {90169,97969,105769}
    {93997,105625,117253}
    {98128,115600,133072}
    {99801,114921,130041}
    {106093,133225,160357}
    {107001,136161,165321}
    {110196,133956,157716}
    {115812,152100,188388}
    {117364,131044,144724}
    {123397,133225,143053}
    {131536,163216,194896}
    {132100,168100,204100}
    {133396,155236,177076}
    {144433,180625,216817}
    {144841,160801,176761}
    {149989,182329,214669}
    {153325,180625,207925}
    {158337,189225,220113}
    {159841,203401,246961}
    {161317,198025,234733}
    {165061,177241,189421}
    {167193,189225,211257}
    {176884,195364,213844}
    {177424,204304,231184}
    {190957,235225,279493}
    {195904,238144,280384}
    {205525,255025,304525}
    {213973,235225,256477}
    {216481,231361,246241}
    {220788,260100,299412}
    {231556,264196,296836}
    {235789,284089,332389}
    {247941,301401,354861}
    {256612,280900,305188}
    {264069,294849,325629}
    {277225,319225,361225}
    {279073,297025,314977}
    {281488,336400,391312}
    {297232,336400,375568}
    {300141,349281,398421}
    {300517,354025,407533}
    {305329,332929,360529}
    {333541,395641,457741}
    {354349,375769,397189}
    {360676,391876,423076}
    {375988,422500,469012}
    {397989,439569,481149}
    {399204,459684,520164}
    {423229,458329,493429}
    {443917,469225,494533}
    {464425,525625,586825}
    {469456,524176,578896}
    {493588,532900,572212}
    {521001,594441,667881}
    {549481,579121,608761}
    {572377,616225,660073}
    {577377,632025,686673}
    {579364,643204,707044}
    {660244,708964,757684}
    {672841,707281,741721}
    {707536,781456,855376}
    {733525,819025,904525}
    {757861,811801,865741}
    {811521,881721,951921}
    {815893,855625,895357}
    {855892,940900,1025908}
    {865924,925444,984964}
    {980629,1026169,1071709}
    {985153,1050625,1116097}
    {1026448,1123600,1220752}
    {1110573,1199025,1287477}
    {1116292,1188100,1259908}
    {1169137,1221025,1272913}
    {1221316,1331716,1442116}
    {1260109,1338649,1417189}
    {1383601,1442401,1501201}
    {1417396,1503076,1588756}
    {1442704,1567504,1692304}
    {1485549,1595169,1704789}
    {1588969,1682209,1775449}
    {1626301,1692601,1758901}
    {1775668,1876900,1978132}
    {1899613,1974025,2048437}
    {1948329,2082249,2216169}
    {1978357,2088025,2197693}
    {2197924,2316484,2435044}
    {2206009,2289169,2372329}
    {2435281,2563201,2691121}
    {2511657,2673225,2834793}
    {2548057,2640625,2733193}
    {2691364,2829124,2966884}
    {2928421,3031081,3133741}
    {2967133,3115225,3263317}
    {3263572,3422500,3581428}
    {3349861,3463321,3576781}
    {3581689,3751969,3922249}
    {3815233,3940225,4065217}
    {3922516,4104676,4286836}
    {4287109,4481689,4676269}
    {4327489,4464769,4602049}
    {4676548,4884100,5091652}
    {4889677,5040025,5190373}
    {5091937,5313025,5534113}
    {5504941,5669161,5833381}
    {5534404,5769604,6004804}
    {6005101,6255001,6504901}
    {6176521,6355441,6534361}
    {6907753,7102225,7296697}
    {7702069,7912969,8123869}
    {8562997,8791225,9019453}
    {9494161,9740641,9987121}
    {10499281,10764961,11030641}
    {11582173,11868025,12153877}
    {12746749,13053769,13360789}
    {13997017,14326225,14655433}
    {15337081,15689521,16041961}
    {16771141,17147881,17524621}
    {18303493,18705625,19107757}
    {19938529,20367169,20795809}
    {21680737,22137025,22593313}
    {23534701,24019801,24504901}

    Paul.

  2. paul says:

    Here are the worked examples

    AP = 48
    52 + 100 – 148 = 2^2
    52 – 100 + 148 = 10^2
    -52 + 100 + 148 = 14^2

    AP = 60
    109 + 169 – 229 = 7^2
    109 – 169 + 229 = 13^2
    -109 + 169 + 229 = 17^2

    AP = 108
    117 + 225 – 333 = 3^2
    117 – 225 + 333 = 15^2
    -117 + 225 + 333 = 21^2

    AP = 120
    169 + 289 – 409 = 7^2
    169 – 289 + 409 = 17^2
    -169 + 289 + 409 = 23^2

    AP = 192
    208 + 400 – 592 = 4^2
    208 – 400 + 592 = 20^2
    -208 + 400 + 592 = 28^2

    AP = 300
    325 + 625 – 925 = 5^2
    325 – 625 + 925 = 25^2
    -325 + 625 + 925 = 35^2

    AP = 420
    421 + 841 – 1261 = 1^2
    421 – 841 + 1261 = 29^2
    -421 + 841 + 1261 = 41^2

    AP = 240
    436 + 676 – 916 = 14^2
    436 – 676 + 916 = 26^2
    -436 + 676 + 916 = 34^2

    AP = 168
    457 + 625 – 793 = 17^2
    457 – 625 + 793 = 25^2
    -457 + 625 + 793 = 31^2

    AP = 432
    468 + 900 – 1332 = 6^2
    468 – 900 + 1332 = 30^2
    -468 + 900 + 1332 = 42^2

    AP = 588
    637 + 1225 – 1813 = 7^2
    637 – 1225 + 1813 = 35^2
    -637 + 1225 + 1813 = 49^2

    AP = 480
    676 + 1156 – 1636 = 14^2
    676 – 1156 + 1636 = 34^2
    -676 + 1156 + 1636 = 46^2

    AP = 768
    832 + 1600 – 2368 = 8^2
    832 – 1600 + 2368 = 40^2
    -832 + 1600 + 2368 = 56^2

    AP = 420
    949 + 1369 – 1789 = 23^2
    949 – 1369 + 1789 = 37^2
    -949 + 1369 + 1789 = 47^2

    AP = 540
    981 + 1521 – 2061 = 21^2
    981 – 1521 + 2061 = 39^2
    -981 + 1521 + 2061 = 51^2

    AP = 972
    1053 + 2025 – 2997 = 9^2
    1053 – 2025 + 2997 = 45^2
    -1053 + 2025 + 2997 = 63^2

    AP = 1200
    1300 + 2500 – 3700 = 10^2
    1300 – 2500 + 3700 = 50^2
    -1300 + 2500 + 3700 = 70^2

    AP = 360
    1321 + 1681 – 2041 = 31^2
    1321 – 1681 + 2041 = 41^2
    -1321 + 1681 + 2041 = 49^2

    AP = 1080
    1521 + 2601 – 3681 = 21^2
    1521 – 2601 + 3681 = 51^2
    -1521 + 2601 + 3681 = 69^2

    AP = 1260
    1549 + 2809 – 4069 = 17^2
    1549 – 2809 + 4069 = 53^2
    -1549 + 2809 + 4069 = 73^2

    AP = 1452
    1573 + 3025 – 4477 = 11^2
    1573 – 3025 + 4477 = 55^2
    -1573 + 3025 + 4477 = 77^2

    AP = 1680
    1684 + 3364 – 5044 = 2^2
    1684 – 3364 + 5044 = 58^2
    -1684 + 3364 + 5044 = 82^2

    AP = 960
    1744 + 2704 – 3664 = 28^2
    1744 – 2704 + 3664 = 52^2
    -1744 + 2704 + 3664 = 68^2

    AP = 672
    1828 + 2500 – 3172 = 34^2
    1828 – 2500 + 3172 = 50^2
    -1828 + 2500 + 3172 = 62^2

    AP = 1728
    1872 + 3600 – 5328 = 12^2
    1872 – 3600 + 5328 = 60^2
    -1872 + 3600 + 5328 = 84^2

    AP = 2028
    2197 + 4225 – 6253 = 13^2
    2197 – 4225 + 6253 = 65^2
    -2197 + 4225 + 6253 = 91^2

    AP = 1848
    2377 + 4225 – 6073 = 23^2
    2377 – 4225 + 6073 = 65^2
    -2377 + 4225 + 6073 = 89^2

    AP = 2352
    2548 + 4900 – 7252 = 14^2
    2548 – 4900 + 7252 = 70^2
    -2548 + 4900 + 7252 = 98^2

    AP = 2640
    2689 + 5329 – 7969 = 7^2
    2689 – 5329 + 7969 = 73^2
    -2689 + 5329 + 7969 = 103^2

    AP = 1920
    2704 + 4624 – 6544 = 28^2
    2704 – 4624 + 6544 = 68^2
    -2704 + 4624 + 6544 = 92^2

    AP = 1500
    2725 + 4225 – 5725 = 35^2
    2725 – 4225 + 5725 = 65^2
    -2725 + 4225 + 5725 = 85^2

    AP = 2700
    2925 + 5625 – 8325 = 15^2
    2925 – 5625 + 8325 = 75^2
    -2925 + 5625 + 8325 = 105^2

    AP = 660
    3061 + 3721 – 4381 = 49^2
    3061 – 3721 + 4381 = 61^2
    -3061 + 3721 + 4381 = 71^2

    AP = 1008
    3217 + 4225 – 5233 = 47^2
    3217 – 4225 + 5233 = 65^2
    -3217 + 4225 + 5233 = 79^2

    AP = 3072
    3328 + 6400 – 9472 = 16^2
    3328 – 6400 + 9472 = 80^2
    -3328 + 6400 + 9472 = 112^2

    AP = 3468
    3757 + 7225 – 10693 = 17^2
    3757 – 7225 + 10693 = 85^2
    -3757 + 7225 + 10693 = 119^2

    AP = 3780
    3789 + 7569 – 11349 = 3^2
    3789 – 7569 + 11349 = 87^2
    -3789 + 7569 + 11349 = 123^2

    AP = 1680
    3796 + 5476 – 7156 = 46^2
    3796 – 5476 + 7156 = 74^2
    -3796 + 5476 + 7156 = 94^2

    AP = 2160
    3924 + 6084 – 8244 = 42^2
    3924 – 6084 + 8244 = 78^2
    -3924 + 6084 + 8244 = 102^2

    AP = 1512
    4113 + 5625 – 7137 = 51^2
    4113 – 5625 + 7137 = 75^2
    -4113 + 5625 + 7137 = 93^2

    AP = 3888
    4212 + 8100 – 11988 = 18^2
    4212 – 8100 + 11988 = 90^2
    -4212 + 8100 + 11988 = 126^2

    AP = 3000
    4225 + 7225 – 10225 = 35^2
    4225 – 7225 + 10225 = 85^2
    -4225 + 7225 + 10225 = 115^2

    AP = 2772
    4453 + 7225 – 9997 = 41^2
    4453 – 7225 + 9997 = 85^2
    -4453 + 7225 + 9997 = 113^2

    AP = 4332
    4693 + 9025 – 13357 = 19^2
    4693 – 9025 + 13357 = 95^2
    -4693 + 9025 + 13357 = 133^2

    AP = 4680
    4729 + 9409 – 14089 = 7^2
    4729 – 9409 + 14089 = 97^2
    -4729 + 9409 + 14089 = 137^2

    AP = 3120
    4801 + 7921 – 11041 = 41^2
    4801 – 7921 + 11041 = 89^2
    -4801 + 7921 + 11041 = 119^2

    AP = 4800
    5200 + 10000 – 14800 = 20^2
    5200 – 10000 + 14800 = 100^2
    -5200 + 10000 + 14800 = 140^2

    AP = 1440
    5284 + 6724 – 8164 = 62^2
    5284 – 6724 + 8164 = 82^2
    -5284 + 6724 + 8164 = 98^2

    AP = 2940
    5341 + 8281 – 11221 = 49^2
    5341 – 8281 + 11221 = 91^2
    -5341 + 8281 + 11221 = 119^2

    AP = 5292
    5733 + 11025 – 16317 = 21^2
    5733 – 11025 + 16317 = 105^2
    -5733 + 11025 + 16317 = 147^2

    AP = 4320
    6084 + 10404 – 14724 = 42^2
    6084 – 10404 + 14724 = 102^2
    -6084 + 10404 + 14724 = 138^2

    AP = 1092
    6133 + 7225 – 8317 = 71^2
    6133 – 7225 + 8317 = 85^2
    -6133 + 7225 + 8317 = 97^2

    AP = 5040
    6196 + 11236 – 16276 = 34^2
    6196 – 11236 + 16276 = 106^2
    -6196 + 11236 + 16276 = 146^2

    AP = 5808
    6292 + 12100 – 17908 = 22^2
    6292 – 12100 + 17908 = 110^2
    -6292 + 12100 + 17908 = 154^2

    AP = 5460
    6421 + 11881 – 17341 = 31^2
    6421 – 11881 + 17341 = 109^2
    -6421 + 11881 + 17341 = 151^2

    AP = 6720
    6736 + 13456 – 20176 = 4^2
    6736 – 13456 + 20176 = 116^2
    -6736 + 13456 + 20176 = 164^2

    AP = 6348
    6877 + 13225 – 19573 = 23^2
    6877 – 13225 + 19573 = 115^2
    -6877 + 13225 + 19573 = 161^2

    AP = 3840
    6976 + 10816 – 14656 = 56^2
    6976 – 10816 + 14656 = 104^2
    -6976 + 10816 + 14656 = 136^2

    AP = 2688
    7312 + 10000 – 12688 = 68^2
    7312 – 10000 + 12688 = 100^2
    -7312 + 10000 + 12688 = 124^2

    AP = 6912
    7488 + 14400 – 21312 = 24^2
    7488 – 14400 + 21312 = 120^2
    -7488 + 14400 + 21312 = 168^2

    AP = 7500
    8125 + 15625 – 23125 = 25^2
    8125 – 15625 + 23125 = 125^2
    -8125 + 15625 + 23125 = 175^2

    AP = 1980
    8221 + 10201 – 12181 = 79^2
    8221 – 10201 + 12181 = 101^2
    -8221 + 10201 + 12181 = 119^2

    AP = 5880
    8281 + 14161 – 20041 = 49^2
    8281 – 14161 + 20041 = 119^2
    -8281 + 14161 + 20041 = 161^2

    AP = 3780
    8541 + 12321 – 16101 = 69^2
    8541 – 12321 + 16101 = 111^2
    -8541 + 12321 + 16101 = 141^2

    AP = 8112
    8788 + 16900 – 25012 = 26^2
    8788 – 16900 + 25012 = 130^2
    -8788 + 16900 + 25012 = 182^2

    AP = 4860
    8829 + 13689 – 18549 = 63^2
    8829 – 13689 + 18549 = 117^2
    -8829 + 13689 + 18549 = 153^2

    AP = 8748
    9477 + 18225 – 26973 = 27^2
    9477 – 18225 + 26973 = 135^2
    -9477 + 18225 + 26973 = 189^2

    AP = 7392
    9508 + 16900 – 24292 = 46^2
    9508 – 16900 + 24292 = 130^2
    -9508 + 16900 + 24292 = 178^2

    AP = 9240
    9529 + 18769 – 28009 = 17^2
    9529 – 18769 + 28009 = 137^2
    -9529 + 18769 + 28009 = 193^2

    AP = 9408
    10192 + 19600 – 29008 = 28^2
    10192 – 19600 + 29008 = 140^2
    -10192 + 19600 + 29008 = 196^2

    AP = 5148
    10477 + 15625 – 20773 = 73^2
    10477 – 15625 + 20773 = 125^2
    -10477 + 15625 + 20773 = 161^2

    AP = 10500
    10525 + 21025 – 31525 = 5^2
    10525 – 21025 + 31525 = 145^2
    -10525 + 21025 + 31525 = 205^2

    AP = 10560
    10756 + 21316 – 31876 = 14^2
    10756 – 21316 + 31876 = 146^2
    -10756 + 21316 + 31876 = 206^2

    AP = 7680
    10816 + 18496 – 26176 = 56^2
    10816 – 18496 + 26176 = 136^2
    -10816 + 18496 + 26176 = 184^2

    AP = 6000
    10900 + 16900 – 22900 = 70^2
    10900 – 16900 + 22900 = 130^2
    -10900 + 16900 + 22900 = 170^2

    AP = 10092
    10933 + 21025 – 31117 = 29^2
    10933 – 21025 + 31117 = 145^2
    -10933 + 21025 + 31117 = 203^2

    AP = 1680
    11089 + 12769 – 14449 = 97^2
    11089 – 12769 + 14449 = 113^2
    -11089 + 12769 + 14449 = 127^2

    AP = 4200
    11425 + 15625 – 19825 = 85^2
    11425 – 15625 + 19825 = 125^2
    -11425 + 15625 + 19825 = 155^2

    AP = 10800
    11700 + 22500 – 33300 = 30^2
    11700 – 22500 + 33300 = 150^2
    -11700 + 22500 + 33300 = 210^2

    AP = 3240
    11889 + 15129 – 18369 = 93^2
    11889 – 15129 + 18369 = 123^2
    -11889 + 15129 + 18369 = 147^2

    AP = 2640
    12244 + 14884 – 17524 = 98^2
    12244 – 14884 + 17524 = 122^2
    -12244 + 14884 + 17524 = 142^2

    AP = 11532
    12493 + 24025 – 35557 = 31^2
    12493 – 24025 + 35557 = 155^2
    -12493 + 24025 + 35557 = 217^2

    AP = 4032
    12868 + 16900 – 20932 = 94^2
    12868 – 16900 + 20932 = 130^2
    -12868 + 16900 + 20932 = 158^2

    AP = 7260
    13189 + 20449 – 27709 = 77^2
    13189 – 20449 + 27709 = 143^2
    -13189 + 20449 + 27709 = 187^2

    AP = 12288
    13312 + 25600 – 37888 = 32^2
    13312 – 25600 + 37888 = 160^2
    -13312 + 25600 + 37888 = 224^2

    AP = 11220
    13429 + 24649 – 35869 = 47^2
    13429 – 24649 + 35869 = 157^2
    -13429 + 24649 + 35869 = 217^2

    AP = 9720
    13689 + 23409 – 33129 = 63^2
    13689 – 23409 + 33129 = 153^2
    -13689 + 23409 + 33129 = 207^2

    AP = 11340
    13941 + 25281 – 36621 = 51^2
    13941 – 25281 + 36621 = 159^2
    -13941 + 25281 + 36621 = 219^2

    AP = 13068
    14157 + 27225 – 40293 = 33^2
    14157 – 27225 + 40293 = 165^2
    -14157 + 27225 + 40293 = 231^2

    AP = 7140
    15061 + 22201 – 29341 = 89^2
    15061 – 22201 + 29341 = 149^2
    -15061 + 22201 + 29341 = 191^2

    AP = 6720
    15184 + 21904 – 28624 = 92^2
    15184 – 21904 + 28624 = 148^2
    -15184 + 21904 + 28624 = 188^2

    AP = 8640
    15696 + 24336 – 32976 = 84^2
    15696 – 24336 + 32976 = 156^2
    -15696 + 24336 + 32976 = 204^2

    AP = 6048
    16452 + 22500 – 28548 = 102^2
    16452 – 22500 + 28548 = 150^2
    -16452 + 22500 + 28548 = 186^2

    AP = 12000
    16900 + 28900 – 40900 = 70^2
    16900 – 28900 + 40900 = 170^2
    -16900 + 28900 + 40900 = 230^2

    AP = 3432
    17593 + 21025 – 24457 = 119^2
    17593 – 21025 + 24457 = 145^2
    -17593 + 21025 + 24457 = 167^2

    AP = 11088
    17812 + 28900 – 39988 = 82^2
    17812 – 28900 + 39988 = 170^2
    -17812 + 28900 + 39988 = 226^2

    AP = 10140
    18421 + 28561 – 38701 = 91^2
    18421 – 28561 + 38701 = 169^2
    -18421 + 28561 + 38701 = 221^2

    AP = 2448
    18577 + 21025 – 23473 = 127^2
    18577 – 21025 + 23473 = 145^2
    -18577 + 21025 + 23473 = 161^2

    AP = 12480
    19204 + 31684 – 44164 = 82^2
    19204 – 31684 + 44164 = 178^2
    -19204 + 31684 + 44164 = 238^2

    AP = 14520
    20449 + 34969 – 49489 = 77^2
    20449 – 34969 + 49489 = 187^2
    -20449 + 34969 + 49489 = 253^2

    AP = 5760
    21136 + 26896 – 32656 = 124^2
    21136 – 26896 + 32656 = 164^2
    -21136 + 26896 + 32656 = 196^2

    AP = 15960
    21289 + 37249 – 53209 = 73^2
    21289 – 37249 + 53209 = 193^2
    -21289 + 37249 + 53209 = 263^2

    AP = 8580
    21349 + 29929 – 38509 = 113^2
    21349 – 29929 + 38509 = 173^2
    -21349 + 29929 + 38509 = 217^2

    AP = 11760
    21364 + 33124 – 44884 = 98^2
    21364 – 33124 + 44884 = 182^2
    -21364 + 33124 + 44884 = 238^2

    AP = 16632
    21393 + 38025 – 54657 = 69^2
    21393 – 38025 + 54657 = 195^2
    -21393 + 38025 + 54657 = 267^2

    AP = 8232
    22393 + 30625 – 38857 = 119^2
    22393 – 30625 + 38857 = 175^2
    -22393 + 30625 + 38857 = 217^2

    AP = 10500
    23725 + 34225 – 44725 = 115^2
    23725 – 34225 + 44725 = 185^2
    -23725 + 34225 + 44725 = 235^2

    AP = 10032
    24193 + 34225 – 44257 = 119^2
    24193 – 34225 + 44257 = 185^2
    -24193 + 34225 + 44257 = 233^2

    AP = 17280
    24336 + 41616 – 58896 = 84^2
    24336 – 41616 + 58896 = 204^2
    -24336 + 41616 + 58896 = 276^2

    AP = 13500
    24525 + 38025 – 51525 = 105^2
    24525 – 38025 + 51525 = 195^2
    -24525 + 38025 + 51525 = 255^2

    AP = 4368
    24532 + 28900 – 33268 = 142^2
    24532 – 28900 + 33268 = 170^2
    -24532 + 28900 + 33268 = 194^2

    AP = 15708
    26317 + 42025 – 57733 = 103^2
    26317 – 42025 + 57733 = 205^2
    -26317 + 42025 + 57733 = 271^2

    AP = 5940
    27549 + 33489 – 39429 = 147^2
    27549 – 33489 + 39429 = 183^2
    -27549 + 33489 + 39429 = 213^2

    AP = 15360
    27904 + 43264 – 58624 = 112^2
    27904 – 43264 + 58624 = 208^2
    -27904 + 43264 + 58624 = 272^2

    AP = 9072
    28953 + 38025 – 47097 = 141^2
    28953 – 38025 + 47097 = 195^2
    -28953 + 38025 + 47097 = 237^2

    AP = 10752
    29248 + 40000 – 50752 = 136^2
    29248 – 40000 + 50752 = 200^2
    -29248 + 40000 + 50752 = 248^2

    AP = 3420
    29341 + 32761 – 36181 = 161^2
    29341 – 32761 + 36181 = 181^2
    -29341 + 32761 + 36181 = 199^2

    AP = 17340
    31501 + 48841 – 66181 = 119^2
    31501 – 48841 + 66181 = 221^2
    -31501 + 48841 + 66181 = 289^2

    AP = 7920
    32884 + 40804 – 48724 = 158^2
    32884 – 40804 + 48724 = 202^2
    -32884 + 40804 + 48724 = 238^2

    AP = 9000
    33025 + 42025 – 51025 = 155^2
    33025 – 42025 + 51025 = 205^2
    -33025 + 42025 + 51025 = 245^2

    AP = 5460
    33349 + 38809 – 44269 = 167^2
    33349 – 38809 + 44269 = 197^2
    -33349 + 38809 + 44269 = 223^2

    AP = 15120
    34164 + 49284 – 64404 = 138^2
    34164 – 49284 + 64404 = 222^2
    -34164 + 49284 + 64404 = 282^2

    AP = 19440
    35316 + 54756 – 74196 = 126^2
    35316 – 54756 + 74196 = 234^2
    -35316 + 54756 + 74196 = 306^2

    AP = 13608
    37017 + 50625 – 64233 = 153^2
    37017 – 50625 + 64233 = 225^2
    -37017 + 50625 + 64233 = 279^2

    AP = 13260
    39181 + 52441 – 65701 = 161^2
    39181 – 52441 + 65701 = 229^2
    -39181 + 52441 + 65701 = 281^2

    AP = 21660
    39349 + 61009 – 82669 = 133^2
    39349 – 61009 + 82669 = 247^2
    -39349 + 61009 + 82669 = 323^2

    AP = 20592
    41908 + 62500 – 83092 = 146^2
    41908 – 62500 + 83092 = 250^2
    -41908 + 62500 + 83092 = 322^2

    AP = 24000
    43600 + 67600 – 91600 = 140^2
    43600 – 67600 + 91600 = 260^2
    -43600 + 67600 + 91600 = 340^2

    AP = 4620
    44221 + 48841 – 53461 = 199^2
    44221 – 48841 + 53461 = 221^2
    -44221 + 48841 + 53461 = 241^2

    AP = 6720
    44356 + 51076 – 57796 = 194^2
    44356 – 51076 + 57796 = 226^2
    -44356 + 51076 + 57796 = 254^2

    AP = 16800
    45700 + 62500 – 79300 = 170^2
    45700 – 62500 + 79300 = 250^2
    -45700 + 62500 + 79300 = 310^2

    AP = 20580
    46501 + 67081 – 87661 = 161^2
    46501 – 67081 + 87661 = 259^2
    -46501 + 67081 + 87661 = 329^2

    AP = 23712
    46513 + 70225 – 93937 = 151^2
    46513 – 70225 + 93937 = 265^2
    -46513 + 70225 + 93937 = 343^2

    AP = 12960
    47556 + 60516 – 73476 = 186^2
    47556 – 60516 + 73476 = 246^2
    -47556 + 60516 + 73476 = 294^2

    AP = 10560
    48976 + 59536 – 70096 = 196^2
    48976 – 59536 + 70096 = 244^2
    -48976 + 59536 + 70096 = 284^2

    AP = 16128
    51472 + 67600 – 83728 = 188^2
    51472 – 67600 + 83728 = 260^2
    -51472 + 67600 + 83728 = 316^2

    AP = 17940
    54421 + 72361 – 90301 = 191^2
    54421 – 72361 + 90301 = 269^2
    -54421 + 72361 + 90301 = 329^2

    AP = 9828
    55197 + 65025 – 74853 = 213^2
    55197 – 65025 + 74853 = 255^2
    -55197 + 65025 + 74853 = 291^2

    AP = 20328
    55297 + 75625 – 95953 = 187^2
    55297 – 75625 + 95953 = 275^2
    -55297 + 75625 + 95953 = 341^2

    AP = 8160
    57889 + 66049 – 74209 = 223^2
    57889 – 66049 + 74209 = 257^2
    -57889 + 66049 + 74209 = 287^2

    AP = 26880
    60736 + 87616 – 114496 = 184^2
    60736 – 87616 + 114496 = 296^2
    -60736 + 87616 + 114496 = 376^2

    AP = 6072
    64153 + 70225 – 76297 = 241^2
    64153 – 70225 + 76297 = 265^2
    -64153 + 70225 + 76297 = 287^2

    AP = 17640
    64729 + 82369 – 100009 = 217^2
    64729 – 82369 + 100009 = 287^2
    -64729 + 82369 + 100009 = 343^2

    AP = 24192
    65808 + 90000 – 114192 = 204^2
    65808 – 90000 + 114192 = 300^2
    -65808 + 90000 + 114192 = 372^2

    AP = 19380
    66469 + 85849 – 105229 = 217^2
    66469 – 85849 + 105229 = 293^2
    -66469 + 85849 + 105229 = 353^2

    AP = 13728
    70372 + 84100 – 97828 = 238^2
    70372 – 84100 + 97828 = 290^2
    -70372 + 84100 + 97828 = 334^2

    AP = 17820
    73989 + 91809 – 109629 = 237^2
    73989 – 91809 + 109629 = 303^2
    -73989 + 91809 + 109629 = 357^2

    AP = 9792
    74308 + 84100 – 93892 = 254^2
    74308 – 84100 + 93892 = 290^2
    -74308 + 84100 + 93892 = 322^2

    AP = 16500
    76525 + 93025 – 109525 = 245^2
    76525 – 93025 + 109525 = 305^2
    -76525 + 93025 + 109525 = 355^2

    AP = 28392
    77233 + 105625 – 134017 = 221^2
    77233 – 105625 + 134017 = 325^2
    -77233 + 105625 + 134017 = 403^2

    AP = 23100
    77389 + 100489 – 123589 = 233^2
    77389 – 100489 + 123589 = 317^2
    -77389 + 100489 + 123589 = 383^2

    AP = 25200
    80425 + 105625 – 130825 = 235^2
    80425 – 105625 + 130825 = 325^2
    -80425 + 105625 + 130825 = 395^2

    AP = 23040
    84544 + 107584 – 130624 = 248^2
    84544 – 107584 + 130624 = 328^2
    -84544 + 107584 + 130624 = 392^2

    AP = 32928
    89572 + 122500 – 155428 = 238^2
    89572 – 122500 + 155428 = 350^2
    -89572 + 122500 + 155428 = 434^2

    AP = 7800
    90169 + 97969 – 105769 = 287^2
    90169 – 97969 + 105769 = 313^2
    -90169 + 97969 + 105769 = 337^2

    AP = 11628
    93997 + 105625 – 117253 = 287^2
    93997 – 105625 + 117253 = 325^2
    -93997 + 105625 + 117253 = 359^2

    AP = 17472
    98128 + 115600 – 133072 = 284^2
    98128 – 115600 + 133072 = 340^2
    -98128 + 115600 + 133072 = 388^2

    AP = 15120
    99801 + 114921 – 130041 = 291^2
    99801 – 114921 + 130041 = 339^2
    -99801 + 114921 + 130041 = 381^2

    AP = 27132
    106093 + 133225 – 160357 = 281^2
    106093 – 133225 + 160357 = 365^2
    -106093 + 133225 + 160357 = 433^2

    AP = 29160
    107001 + 136161 – 165321 = 279^2
    107001 – 136161 + 165321 = 369^2
    -107001 + 136161 + 165321 = 441^2

    AP = 23760
    110196 + 133956 – 157716 = 294^2
    110196 – 133956 + 157716 = 366^2
    -110196 + 133956 + 157716 = 426^2

    AP = 36288
    115812 + 152100 – 188388 = 282^2
    115812 – 152100 + 188388 = 390^2
    -115812 + 152100 + 188388 = 474^2

    AP = 13680
    117364 + 131044 – 144724 = 322^2
    117364 – 131044 + 144724 = 362^2
    -117364 + 131044 + 144724 = 398^2

    AP = 9828
    123397 + 133225 – 143053 = 337^2
    123397 – 133225 + 143053 = 365^2
    -123397 + 133225 + 143053 = 391^2

    AP = 31680
    131536 + 163216 – 194896 = 316^2
    131536 – 163216 + 194896 = 404^2
    -131536 + 163216 + 194896 = 476^2

    AP = 36000
    132100 + 168100 – 204100 = 310^2
    132100 – 168100 + 204100 = 410^2
    -132100 + 168100 + 204100 = 490^2

    AP = 21840
    133396 + 155236 – 177076 = 334^2
    133396 – 155236 + 177076 = 394^2
    -133396 + 155236 + 177076 = 446^2

    AP = 36192
    144433 + 180625 – 216817 = 329^2
    144433 – 180625 + 216817 = 425^2
    -144433 + 180625 + 216817 = 503^2

    AP = 15960
    144841 + 160801 – 176761 = 359^2
    144841 – 160801 + 176761 = 401^2
    -144841 + 160801 + 176761 = 439^2

    AP = 32340
    149989 + 182329 – 214669 = 343^2
    149989 – 182329 + 214669 = 427^2
    -149989 + 182329 + 214669 = 497^2

    AP = 27300
    153325 + 180625 – 207925 = 355^2
    153325 – 180625 + 207925 = 425^2
    -153325 + 180625 + 207925 = 485^2

    AP = 30888
    158337 + 189225 – 220113 = 357^2
    158337 – 189225 + 220113 = 435^2
    -158337 + 189225 + 220113 = 501^2

    AP = 43560
    159841 + 203401 – 246961 = 341^2
    159841 – 203401 + 246961 = 451^2
    -159841 + 203401 + 246961 = 539^2

    AP = 36708
    161317 + 198025 – 234733 = 353^2
    161317 – 198025 + 234733 = 445^2
    -161317 + 198025 + 234733 = 521^2

    AP = 12180
    165061 + 177241 – 189421 = 391^2
    165061 – 177241 + 189421 = 421^2
    -165061 + 177241 + 189421 = 449^2

    AP = 22032
    167193 + 189225 – 211257 = 381^2
    167193 – 189225 + 211257 = 435^2
    -167193 + 189225 + 211257 = 483^2

    AP = 18480
    176884 + 195364 – 213844 = 398^2
    176884 – 195364 + 213844 = 442^2
    -176884 + 195364 + 213844 = 482^2

    AP = 26880
    177424 + 204304 – 231184 = 388^2
    177424 – 204304 + 231184 = 452^2
    -177424 + 204304 + 231184 = 508^2

    AP = 44268
    190957 + 235225 – 279493 = 383^2
    190957 – 235225 + 279493 = 485^2
    -190957 + 235225 + 279493 = 569^2

    AP = 42240
    195904 + 238144 – 280384 = 392^2
    195904 – 238144 + 280384 = 488^2
    -195904 + 238144 + 280384 = 568^2

    AP = 49500
    205525 + 255025 – 304525 = 395^2
    205525 – 255025 + 304525 = 505^2
    -205525 + 255025 + 304525 = 595^2

    AP = 21252
    213973 + 235225 – 256477 = 439^2
    213973 – 235225 + 256477 = 485^2
    -213973 + 235225 + 256477 = 527^2

    AP = 14880
    216481 + 231361 – 246241 = 449^2
    216481 – 231361 + 246241 = 481^2
    -216481 + 231361 + 246241 = 511^2

    AP = 39312
    220788 + 260100 – 299412 = 426^2
    220788 – 260100 + 299412 = 510^2
    -220788 + 260100 + 299412 = 582^2

    AP = 32640
    231556 + 264196 – 296836 = 446^2
    231556 – 264196 + 296836 = 514^2
    -231556 + 264196 + 296836 = 574^2

    AP = 48300
    235789 + 284089 – 332389 = 433^2
    235789 – 284089 + 332389 = 533^2
    -235789 + 284089 + 332389 = 617^2

    AP = 53460
    247941 + 301401 – 354861 = 441^2
    247941 – 301401 + 354861 = 549^2
    -247941 + 301401 + 354861 = 639^2

    AP = 24288
    256612 + 280900 – 305188 = 482^2
    256612 – 280900 + 305188 = 530^2
    -256612 + 280900 + 305188 = 574^2

    AP = 30780
    264069 + 294849 – 325629 = 483^2
    264069 – 294849 + 325629 = 543^2
    -264069 + 294849 + 325629 = 597^2

    AP = 42000
    277225 + 319225 – 361225 = 485^2
    277225 – 319225 + 361225 = 565^2
    -277225 + 319225 + 361225 = 635^2

    AP = 17952
    279073 + 297025 – 314977 = 511^2
    279073 – 297025 + 314977 = 545^2
    -279073 + 297025 + 314977 = 577^2

    AP = 54912
    281488 + 336400 – 391312 = 476^2
    281488 – 336400 + 391312 = 580^2
    -281488 + 336400 + 391312 = 668^2

    AP = 39168
    297232 + 336400 – 375568 = 508^2
    297232 – 336400 + 375568 = 580^2
    -297232 + 336400 + 375568 = 644^2

    AP = 49140
    300141 + 349281 – 398421 = 501^2
    300141 – 349281 + 398421 = 591^2
    -300141 + 349281 + 398421 = 669^2

    AP = 53508
    300517 + 354025 – 407533 = 497^2
    300517 – 354025 + 407533 = 595^2
    -300517 + 354025 + 407533 = 679^2

    AP = 27600
    305329 + 332929 – 360529 = 527^2
    305329 – 332929 + 360529 = 577^2
    -305329 + 332929 + 360529 = 623^2

    AP = 62100
    333541 + 395641 – 457741 = 521^2
    333541 – 395641 + 457741 = 629^2
    -333541 + 395641 + 457741 = 721^2

    AP = 21420
    354349 + 375769 – 397189 = 577^2
    354349 – 375769 + 397189 = 613^2
    -354349 + 375769 + 397189 = 647^2

    AP = 31200
    360676 + 391876 – 423076 = 574^2
    360676 – 391876 + 423076 = 626^2
    -360676 + 391876 + 423076 = 674^2

    AP = 46512
    375988 + 422500 – 469012 = 574^2
    375988 – 422500 + 469012 = 650^2
    -375988 + 422500 + 469012 = 718^2

    AP = 41580
    397989 + 439569 – 481149 = 597^2
    397989 – 439569 + 481149 = 663^2
    -397989 + 439569 + 481149 = 723^2

    AP = 60480
    399204 + 459684 – 520164 = 582^2
    399204 – 459684 + 520164 = 678^2
    -399204 + 459684 + 520164 = 762^2

    AP = 35100
    423229 + 458329 – 493429 = 623^2
    423229 – 458329 + 493429 = 677^2
    -423229 + 458329 + 493429 = 727^2

    AP = 25308
    443917 + 469225 – 494533 = 647^2
    443917 – 469225 + 494533 = 685^2
    -443917 + 469225 + 494533 = 721^2

    AP = 61200
    464425 + 525625 – 586825 = 635^2
    464425 – 525625 + 586825 = 725^2
    -464425 + 525625 + 586825 = 805^2

    AP = 54720
    469456 + 524176 – 578896 = 644^2
    469456 – 524176 + 578896 = 724^2
    -469456 + 524176 + 578896 = 796^2

    AP = 39312
    493588 + 532900 – 572212 = 674^2
    493588 – 532900 + 572212 = 730^2
    -493588 + 532900 + 572212 = 782^2

    AP = 73440
    521001 + 594441 – 667881 = 669^2
    521001 – 594441 + 667881 = 771^2
    -521001 + 594441 + 667881 = 861^2

    AP = 29640
    549481 + 579121 – 608761 = 721^2
    549481 – 579121 + 608761 = 761^2
    -549481 + 579121 + 608761 = 799^2

    AP = 43848
    572377 + 616225 – 660073 = 727^2
    572377 – 616225 + 660073 = 785^2
    -572377 + 616225 + 660073 = 839^2

    AP = 54648
    577377 + 632025 – 686673 = 723^2
    577377 – 632025 + 686673 = 795^2
    -577377 + 632025 + 686673 = 861^2

    AP = 63840
    579364 + 643204 – 707044 = 718^2
    579364 – 643204 + 707044 = 802^2
    -579364 + 643204 + 707044 = 878^2

    AP = 48720
    660244 + 708964 – 757684 = 782^2
    660244 – 708964 + 757684 = 842^2
    -660244 + 708964 + 757684 = 898^2

    AP = 34440
    672841 + 707281 – 741721 = 799^2
    672841 – 707281 + 741721 = 841^2
    -672841 + 707281 + 741721 = 881^2

    AP = 73920
    707536 + 781456 – 855376 = 796^2
    707536 – 781456 + 855376 = 884^2
    -707536 + 781456 + 855376 = 964^2

    AP = 85500
    733525 + 819025 – 904525 = 805^2
    733525 – 819025 + 904525 = 905^2
    -733525 + 819025 + 904525 = 995^2

    AP = 53940
    757861 + 811801 – 865741 = 839^2
    757861 – 811801 + 865741 = 901^2
    -757861 + 811801 + 865741 = 959^2

    AP = 70200
    811521 + 881721 – 951921 = 861^2
    811521 – 881721 + 951921 = 939^2
    -811521 + 881721 + 951921 = 1011^2

    AP = 39732
    815893 + 855625 – 895357 = 881^2
    815893 – 855625 + 895357 = 925^2
    -815893 + 855625 + 895357 = 967^2

    AP = 85008
    855892 + 940900 – 1025908 = 878^2
    855892 – 940900 + 1025908 = 970^2
    -855892 + 940900 + 1025908 = 1054^2

    AP = 59520
    865924 + 925444 – 984964 = 898^2
    865924 – 925444 + 984964 = 962^2
    -865924 + 925444 + 984964 = 1022^2

    AP = 45540
    980629 + 1026169 – 1071709 = 967^2
    980629 – 1026169 + 1071709 = 1013^2
    -980629 + 1026169 + 1071709 = 1057^2

    AP = 65472
    985153 + 1050625 – 1116097 = 959^2
    985153 – 1050625 + 1116097 = 1025^2
    -985153 + 1050625 + 1116097 = 1087^2

    AP = 97152
    1026448 + 1123600 – 1220752 = 964^2
    1026448 – 1123600 + 1220752 = 1060^2
    -1026448 + 1123600 + 1220752 = 1148^2

    AP = 88452
    1110573 + 1199025 – 1287477 = 1011^2
    1110573 – 1199025 + 1287477 = 1095^2
    -1110573 + 1199025 + 1287477 = 1173^2

    AP = 71808
    1116292 + 1188100 – 1259908 = 1022^2
    1116292 – 1188100 + 1259908 = 1090^2
    -1116292 + 1188100 + 1259908 = 1154^2

    AP = 51888
    1169137 + 1221025 – 1272913 = 1057^2
    1169137 – 1221025 + 1272913 = 1105^2
    -1169137 + 1221025 + 1272913 = 1151^2

    AP = 110400
    1221316 + 1331716 – 1442116 = 1054^2
    1221316 – 1331716 + 1442116 = 1154^2
    -1221316 + 1331716 + 1442116 = 1246^2

    AP = 78540
    1260109 + 1338649 – 1417189 = 1087^2
    1260109 – 1338649 + 1417189 = 1157^2
    -1260109 + 1338649 + 1417189 = 1223^2

    AP = 58800
    1383601 + 1442401 – 1501201 = 1151^2
    1383601 – 1442401 + 1501201 = 1201^2
    -1383601 + 1442401 + 1501201 = 1249^2

    AP = 85680
    1417396 + 1503076 – 1588756 = 1154^2
    1417396 – 1503076 + 1588756 = 1226^2
    -1417396 + 1503076 + 1588756 = 1294^2

    AP = 124800
    1442704 + 1567504 – 1692304 = 1148^2
    1442704 – 1567504 + 1692304 = 1252^2
    -1442704 + 1567504 + 1692304 = 1348^2

    AP = 109620
    1485549 + 1595169 – 1704789 = 1173^2
    1485549 – 1595169 + 1704789 = 1263^2
    -1485549 + 1595169 + 1704789 = 1347^2

    AP = 93240
    1588969 + 1682209 – 1775449 = 1223^2
    1588969 – 1682209 + 1775449 = 1297^2
    -1588969 + 1682209 + 1775449 = 1367^2

    AP = 66300
    1626301 + 1692601 – 1758901 = 1249^2
    1626301 – 1692601 + 1758901 = 1301^2
    -1626301 + 1692601 + 1758901 = 1351^2

    AP = 101232
    1775668 + 1876900 – 1978132 = 1294^2
    1775668 – 1876900 + 1978132 = 1370^2
    -1775668 + 1876900 + 1978132 = 1442^2

    AP = 74412
    1899613 + 1974025 – 2048437 = 1351^2
    1899613 – 1974025 + 2048437 = 1405^2
    -1899613 + 1974025 + 2048437 = 1457^2

    AP = 133920
    1948329 + 2082249 – 2216169 = 1347^2
    1948329 – 2082249 + 2216169 = 1443^2
    -1948329 + 2082249 + 2216169 = 1533^2

    AP = 109668
    1978357 + 2088025 – 2197693 = 1367^2
    1978357 – 2088025 + 2197693 = 1445^2
    -1978357 + 2088025 + 2197693 = 1519^2

    AP = 118560
    2197924 + 2316484 – 2435044 = 1442^2
    2197924 – 2316484 + 2435044 = 1522^2
    -2197924 + 2316484 + 2435044 = 1598^2

    AP = 83160
    2206009 + 2289169 – 2372329 = 1457^2
    2206009 – 2289169 + 2372329 = 1513^2
    -2206009 + 2289169 + 2372329 = 1567^2

    AP = 127920
    2435281 + 2563201 – 2691121 = 1519^2
    2435281 – 2563201 + 2691121 = 1601^2
    -2435281 + 2563201 + 2691121 = 1679^2

    AP = 161568
    2511657 + 2673225 – 2834793 = 1533^2
    2511657 – 2673225 + 2834793 = 1635^2
    -2511657 + 2673225 + 2834793 = 1731^2

    AP = 92568
    2548057 + 2640625 – 2733193 = 1567^2
    2548057 – 2640625 + 2733193 = 1625^2
    -2548057 + 2640625 + 2733193 = 1681^2

    AP = 137760
    2691364 + 2829124 – 2966884 = 1598^2
    2691364 – 2829124 + 2966884 = 1682^2
    -2691364 + 2829124 + 2966884 = 1762^2

    AP = 102660
    2928421 + 3031081 – 3133741 = 1681^2
    2928421 – 3031081 + 3133741 = 1741^2
    -2928421 + 3031081 + 3133741 = 1799^2

    AP = 148092
    2967133 + 3115225 – 3263317 = 1679^2
    2967133 – 3115225 + 3263317 = 1765^2
    -2967133 + 3115225 + 3263317 = 1847^2

    AP = 158928
    3263572 + 3422500 – 3581428 = 1762^2
    3263572 – 3422500 + 3581428 = 1850^2
    -3263572 + 3422500 + 3581428 = 1934^2

    AP = 113460
    3349861 + 3463321 – 3576781 = 1799^2
    3349861 – 3463321 + 3576781 = 1861^2
    -3349861 + 3463321 + 3576781 = 1921^2

    AP = 170280
    3581689 + 3751969 – 3922249 = 1847^2
    3581689 – 3751969 + 3922249 = 1937^2
    -3581689 + 3751969 + 3922249 = 2023^2

    AP = 124992
    3815233 + 3940225 – 4065217 = 1921^2
    3815233 – 3940225 + 4065217 = 1985^2
    -3815233 + 3940225 + 4065217 = 2047^2

    AP = 182160
    3922516 + 4104676 – 4286836 = 1934^2
    3922516 – 4104676 + 4286836 = 2026^2
    -3922516 + 4104676 + 4286836 = 2114^2

    AP = 194580
    4287109 + 4481689 – 4676269 = 2023^2
    4287109 – 4481689 + 4676269 = 2117^2
    -4287109 + 4481689 + 4676269 = 2207^2

    AP = 137280
    4327489 + 4464769 – 4602049 = 2047^2
    4327489 – 4464769 + 4602049 = 2113^2
    -4327489 + 4464769 + 4602049 = 2177^2

    AP = 207552
    4676548 + 4884100 – 5091652 = 2114^2
    4676548 – 4884100 + 5091652 = 2210^2
    -4676548 + 4884100 + 5091652 = 2302^2

    AP = 150348
    4889677 + 5040025 – 5190373 = 2177^2
    4889677 – 5040025 + 5190373 = 2245^2
    -4889677 + 5040025 + 5190373 = 2311^2

    AP = 221088
    5091937 + 5313025 – 5534113 = 2207^2
    5091937 – 5313025 + 5534113 = 2305^2
    -5091937 + 5313025 + 5534113 = 2399^2

    AP = 164220
    5504941 + 5669161 – 5833381 = 2311^2
    5504941 – 5669161 + 5833381 = 2381^2
    -5504941 + 5669161 + 5833381 = 2449^2

    AP = 235200
    5534404 + 5769604 – 6004804 = 2302^2
    5534404 – 5769604 + 6004804 = 2402^2
    -5534404 + 5769604 + 6004804 = 2498^2

    AP = 249900
    6005101 + 6255001 – 6504901 = 2399^2
    6005101 – 6255001 + 6504901 = 2501^2
    -6005101 + 6255001 + 6504901 = 2599^2

    AP = 178920
    6176521 + 6355441 – 6534361 = 2449^2
    6176521 – 6355441 + 6534361 = 2521^2
    -6176521 + 6355441 + 6534361 = 2591^2

    AP = 194472
    6907753 + 7102225 – 7296697 = 2591^2
    6907753 – 7102225 + 7296697 = 2665^2
    -6907753 + 7102225 + 7296697 = 2737^2

    AP = 210900
    7702069 + 7912969 – 8123869 = 2737^2
    7702069 – 7912969 + 8123869 = 2813^2
    -7702069 + 7912969 + 8123869 = 2887^2

    AP = 228228
    8562997 + 8791225 – 9019453 = 2887^2
    8562997 – 8791225 + 9019453 = 2965^2
    -8562997 + 8791225 + 9019453 = 3041^2

    AP = 246480
    9494161 + 9740641 – 9987121 = 3041^2
    9494161 – 9740641 + 9987121 = 3121^2
    -9494161 + 9740641 + 9987121 = 3199^2

    AP = 265680
    10499281 + 10764961 – 11030641 = 3199^2
    10499281 – 10764961 + 11030641 = 3281^2
    -10499281 + 10764961 + 11030641 = 3361^2

    AP = 285852
    11582173 + 11868025 – 12153877 = 3361^2
    11582173 – 11868025 + 12153877 = 3445^2
    -11582173 + 11868025 + 12153877 = 3527^2

    AP = 307020
    12746749 + 13053769 – 13360789 = 3527^2
    12746749 – 13053769 + 13360789 = 3613^2
    -12746749 + 13053769 + 13360789 = 3697^2

    AP = 329208
    13997017 + 14326225 – 14655433 = 3697^2
    13997017 – 14326225 + 14655433 = 3785^2
    -13997017 + 14326225 + 14655433 = 3871^2

    AP = 352440
    15337081 + 15689521 – 16041961 = 3871^2
    15337081 – 15689521 + 16041961 = 3961^2
    -15337081 + 15689521 + 16041961 = 4049^2

    AP = 376740
    16771141 + 17147881 – 17524621 = 4049^2
    16771141 – 17147881 + 17524621 = 4141^2
    -16771141 + 17147881 + 17524621 = 4231^2

    AP = 402132
    18303493 + 18705625 – 19107757 = 4231^2
    18303493 – 18705625 + 19107757 = 4325^2
    -18303493 + 18705625 + 19107757 = 4417^2

    AP = 428640
    19938529 + 20367169 – 20795809 = 4417^2
    19938529 – 20367169 + 20795809 = 4513^2
    -19938529 + 20367169 + 20795809 = 4607^2

    AP = 456288
    21680737 + 22137025 – 22593313 = 4607^2
    21680737 – 22137025 + 22593313 = 4705^2
    -21680737 + 22137025 + 22593313 = 4801^2

    AP = 485100
    23534701 + 24019801 – 24504901 = 4801^2
    23534701 – 24019801 + 24504901 = 4901^2
    -23534701 + 24019801 + 24504901 = 4999^2

    Paul.

    • benvitalis says:

      Giving us a second sequence of three integers in A.P.

      • paul says:

        The solutions for (a, b, c) are directly linked to pythagorean triangles, in fact every PT produces one solution set :- (a, b, c). There are no solutions for (a, b, c) that can’t be derived from a PT. Lets take the first solution for (a, b, c) is (13, 25, 37), this comes from the first PT (3, 4, 5), the resuting x^2, y^2, z^2 is (1, 25, 49) which also happens to be the first set of 3 squares in AP. let us now take the roots of those numbers (1, 5, 7) these are related to the (3, 4, 5) by 1 = 4 – 3, 5 = 5, 7 = 4 + 3. So, (x, y, z) come from the PT, we solve backwards to find (a, b, c) as in {a + b – c = 1}, {a – b + c = 25}, {-a + b + c = 49}, the only solution is (13, 25, 37).

        The full solutions list above was from the set of PT’s generated that have a small leg <=100 (not including this example). The MMa program I wrote generated all the PT's, generated all the (x, y, z's) and then solved the resulting 3 equations for (a, b, c). Stored them all in a list and then sorted the list in to size order by a. and printed the results for each x, y, z. Total time 449 ms

        Paul.

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