## 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   $2 \, k$

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.