A^n + B^n + C^n = D^n + E^n + F^n, n = 2, 6

 
 
A^n \; + \; B^n \; + \; C^n \; = \; D^n \; + \; E^n \; + \; F^n,     n = 2, 6

 

36^2 + 179^2 + 275^2 = 65^2 + 169^2 + 276^2
36^6 + 179^6 + 275^6 = 65^6 + 169^6 + 276^6
 

4807^2 + 17126^2 + 26973^2 = 5094^2 + 17027^2 + 26983^2
4807^6 + 17126^6 + 26973^6 = 5094^6 + 17027^6 + 26983^6
 

483680^2 + 1673839^2 + 2643501^2 = 486521^2 + 1672859^2 + 2643600^2
483680^6 + 1673839^6 + 2643501^6 = 486521^6 + 1672859^6 + 2643600^6
 

47520839^2 + 163976002^2 + 259040463^2 = 47548962^2 + 163966301^2 + 259041443^2
47520839^6 + 163976002^6 + 259040463^6 = 47548962^6 + 163966301^6 + 259041443^6
 

4657795956^2 + 16067547539^2 + 25383364975^2
= 4658074345^2 + 16067451509^2 + 25383374676^2

4657795956^6 + 16067547539^6 + 25383364975^6
= 4658074345^6 + 16067451509^6 + 25383374676^6
 

1574451457526^2 + 456428731967^2 + 2487311153913^2
= 2487311249943^2 + 456431487734^2 + 1574450506927^2

1574451457526^6 + 456428731967^6 + 2487311153913^6
= 2487311249943^6 + 456431487734^6 + 1574450506927^6

 
154280133463679^2+44725479190560^2+243731113943801^2
= 243731114894400^2+44725506469841^2+154280124053719^2

154280133463679^6+44725479190560^6+243731113943801^6
= 243731114894400^6+44725506469841^6+154280124053719^6

 
15117878213944802^2+4382641732231279^2+23883161897164923^2
= 23883161906574883^2+4382642002268322^2+15117878120795801^2

15117878213944802^6+4382641732231279^6+23883161897164923^6
= 23883161906574883^6+4382642002268322^6+15117878120795801^6

 
1481397780734570899^2+429454176161104676^2+2340306135222256875^2
= 2340306135315405876^2+429454178834195825^2+1481397779812490849^2

1481397780734570899^6+429454176161104676^6+2340306135222256875^6
=2340306135315405876^6+429454178834195825^6+1481397779812490849^6

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to A^n + B^n + C^n = D^n + E^n + F^n, n = 2, 6

  1. Paul says:

    Here is a list of solutions with a < b < c < 276

    1^2 + 132^2 + 133^2 = 71^2 + 92^2 + 147^2 = 35114
    1^6 + 132^6 + 133^6 = 71^6 + 92^6 + 147^6 = 10824753654794

    3^2 + 19^2 + 22^2 = 10^2 + 15^2 + 23^2 = 854
    3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6 = 160426514

    3^2 + 55^2 + 80^2 = 32^2 + 43^2 + 81^2 = 9434
    3^6 + 55^6 + 80^6 = 32^6 + 43^6 + 81^6 = 289824641354

    6^2 + 38^2 + 44^2 = 20^2 + 30^2 + 46^2 = 3416
    6^6 + 38^6 + 44^6 = 20^6 + 30^6 + 46^6 = 10267296896

    6^2 + 110^2 + 160^2 = 64^2 + 86^2 + 162^2 = 37736
    6^6 + 110^6 + 160^6 = 64^6 + 86^6 + 162^6 = 18548777046656

    9^2 + 57^2 + 66^2 = 30^2 + 45^2 + 69^2 = 7686
    9^6 + 57^6 + 66^6 = 30^6 + 45^6 + 69^6 = 116950928706

    9^2 + 165^2 + 240^2 = 96^2 + 129^2 + 243^2 = 84906
    9^6 + 165^6 + 240^6 = 96^6 + 129^6 + 243^6 = 211282163547066

    11^2 + 65^2 + 78^2 = 37^2 + 50^2 + 81^2 = 10430
    11^6 + 65^6 + 78^6 = 37^6 + 50^6 + 81^6 = 300620262890

    11^2 + 188^2 + 243^2 = 103^2 + 148^2 + 249^2 = 94514
    11^6 + 188^6 + 243^6 = 103^6 + 148^6 + 249^6 = 250042799853794

    12^2 + 76^2 + 88^2 = 40^2 + 60^2 + 92^2 = 13664
    12^6 + 76^6 + 88^6 = 40^6 + 60^6 + 92^6 = 657107001344

    14^2 + 163^2 + 243^2 = 75^2 + 142^2 + 245^2 = 85814
    14^6 + 163^6 + 243^6 = 75^6 + 142^6 + 245^6 = 224646509202194

    15^2 + 52^2 + 65^2 = 36^2 + 37^2 + 67^2 = 7154
    15^6 + 52^6 + 65^6 = 36^6 + 37^6 + 67^6 = 95200890914

    15^2 + 95^2 + 110^2 = 50^2 + 75^2 + 115^2 = 21350
    15^6 + 95^6 + 110^6 = 50^6 + 75^6 + 115^6 = 2506664281250

    18^2 + 114^2 + 132^2 = 60^2 + 90^2 + 138^2 = 30744
    18^6 + 114^6 + 132^6 = 60^6 + 90^6 + 138^6 = 7484859437184

    21^2 + 133^2 + 154^2 = 70^2 + 105^2 + 161^2 = 41846
    21^6 + 133^6 + 154^6 = 70^6 + 105^6 + 161^6 = 18874018945586

    22^2 + 130^2 + 156^2 = 74^2 + 100^2 + 162^2 = 41720
    22^6 + 130^6 + 156^6 = 74^6 + 100^6 + 162^6 = 19239696824960

    23^2 + 54^2 + 73^2 = 33^2 + 47^2 + 74^2 = 8774
    23^6 + 54^6 + 73^6 = 33^6 + 47^6 + 74^6 = 176277173474

    24^2 + 152^2 + 176^2 = 80^2 + 120^2 + 184^2 = 54656
    24^6 + 152^6 + 176^6 = 80^6 + 120^6 + 184^6 = 42054848086016

    26^2 + 169^2 + 225^2 = 111^2 + 121^2 + 230^2 = 79862
    26^6 + 169^6 + 225^6 = 111^6 + 121^6 + 230^6 = 153044731928882

    27^2 + 171^2 + 198^2 = 90^2 + 135^2 + 207^2 = 69174
    27^6 + 171^6 + 198^6 = 90^6 + 135^6 + 207^6 = 85257227026674

    29^2 + 197^2 + 261^2 = 131^2 + 139^2 + 267^2 = 107771
    29^6 + 197^6 + 261^6 = 131^6 + 139^6 + 267^6 = 374565823668011

    30^2 + 104^2 + 130^2 = 72^2 + 74^2 + 134^2 = 28616
    30^6 + 104^6 + 130^6 = 72^6 + 74^6 + 134^6 = 6092857018496

    30^2 + 190^2 + 220^2 = 100^2 + 150^2 + 230^2 = 85400
    30^6 + 190^6 + 220^6 = 100^6 + 150^6 + 230^6 = 160426514000000

    33^2 + 195^2 + 234^2 = 111^2 + 150^2 + 243^2 = 93870
    33^6 + 195^6 + 234^6 = 111^6 + 150^6 + 243^6 = 219152171646810

    33^2 + 209^2 + 242^2 = 110^2 + 165^2 + 253^2 = 103334
    33^6 + 209^6 + 242^6 = 110^6 + 165^6 + 253^6 = 284205355568354

    36^2 + 179^2 + 275^2 = 65^2 + 169^2 + 276^2 = 108962
    36^6 + 179^6 + 275^6 = 65^6 + 169^6 + 276^6 = 465406299992882

    36^2 + 228^2 + 264^2 = 120^2 + 180^2 + 276^2 = 122976
    36^6 + 228^6 + 264^6 = 120^6 + 180^6 + 276^6 = 479031003979776

    40^2 + 125^2 + 129^2 = 51^2 + 113^2 + 136^2 = 33866
    40^6 + 125^6 + 129^6 = 51^6 + 113^6 + 136^6 = 8427066928346

    45^2 + 156^2 + 195^2 = 108^2 + 111^2 + 201^2 = 64386
    45^6 + 156^6 + 195^6 = 108^6 + 111^6 + 201^6 = 69401449476306

    46^2 + 108^2 + 146^2 = 66^2 + 94^2 + 148^2 = 35096
    46^6 + 108^6 + 146^6 = 66^6 + 94^6 + 148^6 = 11281739102336

    60^2 + 208^2 + 260^2 = 144^2 + 148^2 + 268^2 = 114464
    60^6 + 208^6 + 260^6 = 144^6 + 148^6 + 268^6 = 389942849183744

    69^2 + 162^2 + 219^2 = 99^2 + 141^2 + 222^2 = 78966
    69^6 + 162^6 + 219^6 = 99^6 + 141^6 + 222^6 = 128506059462546

    80^2 + 250^2 + 258^2 = 102^2 + 226^2 + 272^2 = 135464
    80^6 + 250^6 + 258^6 = 102^6 + 226^6 + 272^6 = 539332283414144

    Paul.

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