a^2 – 1, b^2 – 1, c^2 – 1 in arithmetical progression

 
 
We want    a^2 - 1,    b^2 - 1,    c^2 - 1   to be in arithmetical progression

and the sum    (a^2 - 1) + (b^2 - 1) + (c^2 - 1)    to be a square number

 
Here are the first few solutions:

(14, 26, 34)
(26^2 - 1) - (14^2 - 1) = 480 = (34^2 - 1) - (26^2 - 1)
(14^2 - 1) + (26^2 - 1) + (34^2 - 1) = 45^2

(322, 362, 398)
(362^2 - 1) - (322^2 - 1) = 27360 = (398^2 - 1) - (362^2 - 1)
(322^2 - 1) + (362^2 - 1) + (398^2 - 1) = 627^2

(4898, 5042, 5182)
(5042^2 - 1) - (4898^2 - 1) = 1431360 = (5182^2 - 1) - (5042^2 - 1)
(4898^2 - 1) + (5042^2 - 1) + (5182^2 - 1) = 8733^2

(69694, 70226, 70754)
(70226^2 - 1) - (69694^2 - 1) = 74437440 = (70754^2 - 1) - (70226^2 - 1)
(69694^2 - 1) + (70226^2 - 1) + (70754^2 - 1) = 121635^2

(976142, 978122, 980098)
(978122^2 - 1) - (976142^2 - 1) = 3869442720 = (980098^2 - 1) - (978122^2 - 1)
(976142^2 - 1) + (978122^2 - 1) + (980098^2 - 1) = 1694157^2

(13616098, 13623482, 13630862)
(13623482^2 - 1) - (13616098^2 - 1) = 201137058720
= (13630862^2 - 1) - (13623482^2 - 1)
(13616098^2 - 1) + (13623482^2 - 1) + (13630862^2 - 1) = 23596563^2

(189723074, 189750626, 189778174)
(189750626^2 - 1) - (189723074^2 - 1) = 10455259382400
= (189778174^2 - 1) - (189750626^2 - 1)
(189723074^2 - 1) + (189750626^2 - 1) + (189778174^2 - 1) = 328657725^2

(2642782462, 2642885282, 2642988098)
(2642885282^2 - 1) - (2642782462^2 - 1) = 543472357438080
= (2642988098^2 - 1) - (2642885282^2 - 1)
(2642782462^2 - 1) + (2642885282^2 - 1) + (2642988098^2 - 1) = 4577611587^2

the next set of solutions are:

(36810259598, 36810643322, 36811027042)
(36810259598^2 - 1) + (36810643322^2 - 1) + (36811027042^2 - 1) = 63757904493^2

(512704689154, 512706121226, 512707553294)
(512704689154^2 - 1) + (512706121226^2 - 1) + (512707553294^2 - 1) = 888033051315^2

(7141069709282, 7141075053842, 7141080398398)
(7141069709282^2 - 1) + (7141075053842^2 - 1) + (7141080398398^2 - 1)
= 12368704813917^2

(99462324686398, 99462344632562, 99462364578722)
(99462324686398^2 - 1) + (99462344632562^2 - 1) + (99462364578722^2 - 1)
= 172273834343523^2

(1385331675361934, 1385331749802026, 1385331824242114)
(1385331675361934^2 - 1) + (1385331749802026^2 - 1) + (1385331824242114^2 - 1)
= 2399464975995405^2

(19295181874781602, 19295182152595802, 19295182430409998)
(19295181874781602^2 - 1) + (19295182152595802^2 - 1) + (19295182430409998^2 - 1)
= 33420235829592147^2

 
Determine the recursion formula
 
 
 
 

 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to a^2 – 1, b^2 – 1, c^2 – 1 in arithmetical progression

  1. paul says:

    The main recurrence is {14, -1}. which is the sum of the 3 terms is a square, some of the other values for {a, b, c} follow that recurrence but some as shown don’t. There are in fact for some values of the sum of 3 many solutions as can be seen from this following list.
    The format is {a, b, c, sqrt of the sum of 3} it is evident what they are by looking at the first few for reference. It can be seen in the 4th solution there are 13 versions that work. the last one in this list has 364 solutions.

    {14,26,34,45}

    {322,362,398,627}

    {4898,5042,5182,8733}

    {334,70226,99314,121635}
    {5986,70226,99134,121635}
    {6734,70226,99086,121635}
    {13106,70226,98446,121635}
    {25766,70226,95914,121635}
    {31894,70226,94054,121635}
    {37814,70226,91834,121635}
    {38506,70226,91546,121635}
    {43654,70226,89206,121635}
    {44326,70226,88874,121635}
    {60674,70226,78626,121635}
    {65554,70226,74606,121635}
    {69694,70226,70754,121635}

    {976142,978122,980098,1694157}

    {13616098,13623482,13630862,23596563}

    {1161746,189750626,268345394,328657725}
    {6623014,189750626,268266166,328657725}
    {7465094,189750626,268244054,328657725}
    {21574114,189750626,267479266,328657725}
    {27013526,189750626,266984774,328657725}
    {27890234,189750626,266894614,328657725}
    {35582066,189750626,265978414,328657725}
    {41861426,189750626,265062674,328657725}
    {49532854,189750626,263736794,328657725}
    {54890306,189750626,262674046,328657725}
    {55714594,189750626,262500446,328657725}
    {55752706,189750626,262492354,328657725}
    {63305866,189750626,260773786,328657725}
    {69454406,189750626,259203946,328657725}
    {75561554,189750626,257489906,328657725}
    {82962074,189750626,255201674,328657725}
    {82999126,189750626,255189626,328657725}
    {96279934,189750626,250481086,328657725}
    {102137386,189750626,248150266,328657725}
    {102173414,189750626,248135434,328657725}
    {109292846,189750626,245082994,328657725}
    {110061646,189750626,244738706,328657725}
    {120755446,189750626,239642906,328657725}
    {127622014,189750626,236057666,328657725}
    {127656286,189750626,236039134,328657725}
    {128396606,189750626,235637246,328657725}
    {135144374,189750626,231833126,328657725}
    {145244174,189750626,225642926,328657725}
    {145984466,189750626,225164686,328657725}
    {151696454,189750626,221356694,328657725}
    {152390566,189750626,220879414,328657725}
    {152422634,189750626,220857286,328657725}
    {158733794,189750626,216365854,328657725}
    {168127994,189750626,209149654,328657725}
    {168783706,189750626,208620854,328657725}
    {168813994,189750626,208596346,328657725}
    {174734866,189750626,203662286,328657725}
    {174764434,189750626,203636914,328657725}
    {184194746,189750626,195148394,328657725}
    {189723074,189750626,189778174,328657725}

    {2642782462,2642885282,2642988098,4577611587}

    {36810259598,36810643322,36811027042,63757904493}

    {3035789266,512706121226,725069594914,888033051315}
    {17385859694,512706121226,724867481246,888033051315}
    {37793715826,512706121226,724090304126,888033051315}
    {65704881086,512706121226,722092793266,888033051315}
    {86013414934,512706121226,719956127786,888033051315}
    {86015425886,512706121226,719955887534,888033051315}
    {106255728166,512706121226,717248111686,888033051315}
    {106257731554,512706121226,717247814894,888033051315}
    {126413744906,512706121226,713971076854,888033051315}
    {157914649646,512706121226,707670895906,888033051315}
    {167864229106,512706121226,705377015554,888033051315}
    {177780484294,512706121226,702943264346,888033051315}
    {177782447726,512706121226,702942767774,888033051315}
    {187663476974,512706121226,700369583074,888033051315}
    {197507255926,512706121226,697657521526,888033051315}
    {197509204594,512706121226,697656969854,888033051315}
    {207313844654,512706121226,694806522206,888033051315}
    {217077337946,512706121226,691818301894,888033051315}
    {257080514146,512706121226,677971048594,888033051315}
    {276070654106,512706121226,670462621946,888033051315}
    {276072526814,512706121226,670461850834,888033051315}
    {294843670966,512706121226,662421575114,888033051315}
    {294845521214,512706121226,662420751566,888033051315}
    {313382777734,512706121226,653855005414,888033051315}
    {338378656874,512706121226,641276085674,888033051315}
    {342037365586,512706121226,639332131234,888033051315}
    {356304965254,512706121226,631491809306,888033051315}
    {359907474254,512706121226,629445584194,888033051315}
    {373948604086,512706121226,621206547766,888033051315}
    {377492055374,512706121226,619059675326,888033051315}
    {417948482714,512706121226,592498269434,888033051315}
    {426250448234,512706121226,586554080086,888033051315}
    {434469477814,512706121226,580492382666,888033051315}
    {442600741466,512706121226,574316739386,888033051315}
    {450645792646,512706121226,568025970406,888033051315}
    {458599903606,512706121226,561623772554,888033051315}
    {483026628946,512706121226,540759104606,888033051315}
    {498063185594,512706121226,526942308646,888033051315}
    {498064657426,512706121226,526940917474,888033051315}
    {512704689154,512706121226,512707553294,888033051315}

    {186009519022,7141075053842,10097292028462,12368704813917}
    {450586455298,7141075053842,10088948294818,12368704813917}
    {714853823182,7141075053842,10073673106702,12368704813917}
    {3282682652278,7141075053842,9550596863738,12368704813917}
    {3531867445402,7141075053842,9461279944982,12368704813917}
    {3531874526458,7141075053842,9461277301642,12368704813917}
    {3778625724118,7141075053842,9365462801498,12368704813917}
    {3778632733462,7141075053842,9365459973478,12368704813917}
    {4022794889818,7141075053842,9263208252202,12368704813917}
    {6757065224642,7141075053842,7505463036958,12368704813917}
    {6951455411438,7141075053842,7325788251922,12368704813917}
    {6951460894238,7141075053842,7325783049278,12368704813917}
    {7141069709282,7141075053842,7141080398398,12368704813917}

    {76504100818322,99462344632562,118036598381998,172273834343523}
    {88472008378762,99462344632562,109353645266362,172273834343523}
    {88472030308522,99462344632562,109353627524198,172273834343523}
    {99462324686398,99462344632562,99462364578722,172273834343523}

    {3205214804254,1385331749802026,1959152327058094,2399464975995405}
    {5443827442034,1385331749802026,1959147385665986,2399464975995405}
    {6588739276534,1385331749802026,1959143869789514,2399464975995405}
    {8197124015506,1385331749802026,1959137800456354,2399464975995405}
    {9342029778326,1385331749802026,1959132675573226,2399464975995405}
    {9342135051314,1385331749802026,1959132675071234,2399464975995405}
    {10598114100806,1385331749802026,1959126283320346,2399464975995405}
    {17990968270366,1385331749802026,1959072341461586,2399464975995405}
    {21374227286174,1385331749802026,1959038349912274,2399464975995405}
    {24127460160154,1385331749802026,1959006375611194,2399464975995405}
    {25383460467274,1385331749802026,1958990503793674,2399464975995405}
    {26880645378034,1385331749802026,1958970531918014,2399464975995405}
    {28136516874986,1385331749802026,1958952896431466,2399464975995405}
    {36784478540626,1385331749802026,1958809591603426,2399464975995405}
    {42920055195194,1385331749802026,1958684758423654,2399464975995405}
    {83404575919834,1385331749802026,1957378806142214,2399464975995405}
    {98174660476886,1385331749802026,1956693601476566,2399464975995405}
    {99429057604994,1385331749802026,1956630260555854,2399464975995405}
    {102178829037574,1385331749802026,1956488589518374,2399464975995405}
    {103322170570946,1385331749802026,1956428542801294,2399464975995405}
    {111958150130834,1385331749802026,1955953344698786,2399464975995405}
    {114193036472926,1385331749802026,1955824139446126,2399464975995405}
    {115335987654554,1385331749802026,1955757071819206,2399464975995405}
    {116941555306814,1385331749802026,1955661726030734,2399464975995405}
    {118084411105786,1385331749802026,1955593052215334,2399464975995405}
    {118084516188574,1385331749802026,1955593045870126,2399464975995405}
    {119338203669706,1385331749802026,1955516941160054,2399464975995405}
    {126716558396794,1385331749802026,1955052691833946,2399464975995405}
    {126716663450546,1385331749802026,1955052685024894,2399464975995405}
    {129464085748114,1385331749802026,1954872672201566,2399464975995405}
    {130092818449714,1385331749802026,1954830931975166,2399464975995405}
    {132839925771026,1385331749802026,1954646174666974,2399464975995405}
    {132840030802934,1385331749802026,1954646167528886,2399464975995405}
    {134093213244614,1385331749802026,1954560596190566,2399464975995405}
    {135586980773054,1385331749802026,1954457542302706,2399464975995405}
    {136839936697126,1385331749802026,1954370217165574,2399464975995405}
    {141467854121386,1385331749802026,1954040675182934,2399464975995405}
    {144213944605166,1385331749802026,1953839924917214,2399464975995405}
    {145466607791486,1385331749802026,1953747061427534,2399464975995405}
    {147587979094894,1385331749802026,1953587956158046,2399464975995405}
    {150333427278214,1385331749802026,1953378605048666,2399464975995405}
    {151585793434294,1385331749802026,1953281818183946,2399464975995405}
    {160207410928726,1385331749802026,1952593582777174,2399464975995405}
    {161348466337106,1385331749802026,1952499625206046,2399464975995405}
    {167463666606614,1385331749802026,1951984639894934,2399464975995405}
    {220365802033726,1385331749802026,1946722123805326,2399464975995405}
    {225325334153074,1385331749802026,1946154312434626,2399464975995405}
    {226462615375094,1385331749802026,1946022301479754,2399464975995405}
    {235051454846486,1385331749802026,1945003580354966,2399464975995405}
    {235051559360254,1385331749802026,1945003567724594,2399464975995405}
    {237784752075806,1385331749802026,1944671315595346,2399464975995405}
    {238410198145406,1385331749802026,1944594737069746,2399464975995405}
    {241142812991134,1385331749802026,1944257765256914,2399464975995405}
    {241142917464826,1385331749802026,1944257752299226,2399464975995405}
    {243875160483346,1385331749802026,1943916927267806,2399464975995405}
    {245121336668234,1385331749802026,1943760181794314,2399464975995405}
    {249723823978694,1385331749802026,1943174239680154,2399464975995405}
    {252454431345766,1385331749802026,1942821369584186,2399464975995405}
    {252454535742274,1385331749802026,1942821356018674,2399464975995405}
    {253700112191794,1385331749802026,1942659096983554,2399464975995405}
    {255809379931586,1385331749802026,1942382473962866,2399464975995405}
    {258538972928906,1385331749802026,1942021038376246,2399464975995405}
    {259784035008986,1385331749802026,1941854878504966,2399464975995405}
    {261028990812206,1385331749802026,1941687920334946,2399464975995405}
    {267109835569514,1385331749802026,1940860749709034,2399464975995405}
    {268244016133486,1385331749802026,1940704321072034,2399464975995405}
    {268354151975834,1385331749802026,1940689094918714,2399464975995405}
    {269488232013406,1385331749802026,1940531939140754,2399464975995405}
    {272215106486654,1385331749802026,1940151295600306,2399464975995405}
    {273188059472866,1385331749802026,1940014535559086,2399464975995405}
    {275565414285274,1385331749802026,1939678276536826,2399464975995405}
    {280777685747494,1385331749802026,1938930634449146,2399464975995405}
    {284125768487846,1385331749802026,1938442844579194,2399464975995405}
    {286849796721826,1385331749802026,1938041616720274,2399464975995405}
    {292919093155754,1385331749802026,1937133583128406,2399464975995405}
    {301468148290594,1385331749802026,1935821549003246,2399464975995405}
    {342661443185626,1385331749802026,1928955999854074,2399464975995405}
    {345372080694706,1385331749802026,1928472514685054,2399464975995405}
    {348702006107954,1385331749802026,1927873186948606,2399464975995405}
    {357209635550794,1385331749802026,1926314976915446,2399464975995405}
    {359916443487146,1385331749802026,1925811067505494,2399464975995405}
    {359916546969614,1385331749802026,1925811048165566,2399464975995405}
    {363241854384526,1385331749802026,1925186606342974,2399464975995405}
    {365947174021606,1385331749802026,1924674200960954,2399464975995405}
    {368414859958306,1385331749802026,1924203368924654,2399464975995405}
    {374440409438374,1385331749802026,1923039909569926,2399464975995405}
    {375564157293794,1385331749802026,1922820760699054,2399464975995405}
    {375673278059254,1385331749802026,1922799444084406,2399464975995405}
    {376796885177714,1385331749802026,1922579574774334,2399464975995405}
    {379498424196946,1385331749802026,1922048141970206,2399464975995405}
    {380462284948046,1385331749802026,1921857581547394,2399464975995405}
    {382817307178166,1385331749802026,1921389867607286,2399464975995405}
    {384049116668786,1385331749802026,1921144031561666,2399464975995405}
    {387980037566186,1385331749802026,1920354031023766,2399464975995405}
    {390065006407834,1385331749802026,1919931614614214,2399464975995405}
    {391295878952554,1385331749802026,1919681132150294,2399464975995405}
    {392762914131614,1385331749802026,1919381516869934,2399464975995405}
    {393993330306506,1385331749802026,1919129325942646,2399464975995405}
    {393993433429934,1385331749802026,1919129304771614,2399464975995405}
    {400002963471046,1385331749802026,1917885748222106,2399464975995405}
    {402461916657346,1385331749802026,1917371252433806,2399464975995405}
    {405156213776966,1385331749802026,1916803734464486,2399464975995405}
    {408465898245574,1385331749802026,1916101177910374,2399464975995405}
    {408466001206286,1385331749802026,1916101155961666,2399464975995405}
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    {1293168306987754,1385331749802026,1471734977440406,2399464975995405}
    {1295235343360586,1385331749802026,1469916160646134,2399464975995405}
    {1295235422345774,1385331749802026,1469916091047326,2399464975995405}
    {1297772011854766,1385331749802026,1467677048694814,2399464975995405}
    {1299833419608646,1385331749802026,1465851696211994,2399464975995405}
    {1301711995137346,1385331749802026,1464183730183694,2399464975995405}
    {1304238675294914,1385331749802026,1461933511444366,2399464975995405}
    {1306292004632134,1385331749802026,1460099076314086,2399464975995405}
    {1306292083089806,1385331749802026,1460099006121154,2399464975995405}
    {1307227869969814,1385331749802026,1459261254879766,2399464975995405}
    {1308342753806254,1385331749802026,1458261757223906,2399464975995405}
    {1310859196925486,1385331749802026,1456000096104034,2399464975995405}
    {1317274260868246,1385331749802026,1450198826255594,2399464975995405}
    {1320239334795494,1385331749802026,1447499987177146,2399464975995405}
    {1349882662608454,1385331749802026,1419896091694106,2399464975995405}
    {1360560262974026,1385331749802026,1409668005182474,2399464975995405}
    {1364264238847454,1385331749802026,1406083639269394,2399464975995405}
    {1364969219207554,1385331749802026,1405399282992206,2399464975995405}
    {1371977834455766,1385331749802026,1398558163174186,2399464975995405}
    {1372057201361986,1385331749802026,1398480300258034,2399464975995405}
    {1372874229703046,1385331749802026,1397678240310106,2399464975995405}
    {1374837111193894,1385331749802026,1395747481352954,2399464975995405}
    {1374837186193666,1385331749802026,1395747407476786,2399464975995405}
    {1377245568895874,1385331749802026,1393371004785074,2399464975995405}
    {1379202467633354,1385331749802026,1391434032677494,2399464975995405}
    {1380985553027054,1385331749802026,1389664353845794,2399464975995405}
    {1383383421226606,1385331749802026,1387277342096546,2399464975995405}
    {1384443267873146,1385331749802026,1386219662268506,2399464975995405}
    {1385331675361934,1385331749802026,1385331824242114,2399464975995405}

    Paul.

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