OEIS/A003828

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There seem to be more than 19 solutions by Seiji Tomita.

Seiji Tomita's List as of 2017-07-20

           18796760            15365639             2682440            20615673 Noam Elkies, 1986 
             414560              217519               95800              422481 Roger Frye, 1988
  16921802132217020   15075240668820384  128805698258642759   16774794902382238 Seiji Tomita, 03/13/2006
         4480680305         72584786800          1062203384         23661446513
          582665296           260052385           186668000           589845921 Seiji Tomita, 03/13/2006
         3134081336          2448718655           664793200          3393603777 Seiji Tomita, 01/28/2007
  47886740272114976    8813425670440240   56827813308111785   62940516903410601 Seiji Tomita, 8/13/2008
   3579087147375440   14890026433468471   18565945114216720   20249506709579721 Seiji Tomita, 8/13/2008
      4987588419655       2480452675600        502038853976       5062297699257 Seiji Tomita, 5/15/2008
          275156240           219076465           630662624          638523249  Seiji Tomita, 7/16/2017
            2767624             1390400              673865             2813001 Allan MacLeod,1997 
         1670617271           632671960            50237800          1679142729 Seiji Tomita, 03/13/2006
     22495595284040       7592431981391      27239791692640      29999857938609 Seiji Tomita, 03/13/2006
         5821981400         15355831360           140976551         15434547801 Seiji Tomita, 10/24/2007
    130064300991400     440804942580160     514818101299289     573646321871961 Seiji Tomita, 9/15/2008
            8332208             5507880             1705575             8707481 D.J. Bernstein,2001 
       894416022327        125777308440       2032977944240       2051764828361 Seiji Tomita, 7/15/2017
         1166705840           859396455           588903336          1259768473 Robert Gerbicz, ... 2008
        41714673255         18125123544         34169217200         46055390617 Seiji Tomita, 7/16/2017
1171431921806295280 5113654767499398206 4048310673060768880 5557992180974240706 Andrew Bremner, 2013
           11289040             8282543             5870000            12197457 D.J. Bernstein,2001 
         1662997663          1237796960           686398000          1787882337 Robert Gerbicz, ...
  54772209880944552   50049879917257109   16178554328069755   62586521087452988 Andrew Bremner, 2013
          338211440           403510290           572637088           953161234
        17111129720         52289667920         55479193841         64244765937 Seiji Tomita, 7/20/2017
     18776929334105      23449050222680      12035933588696      25866132798297 Seiji Tomita, 7/20/2017
           27450160           146627384           108644015           156646737 Juergen Rathmann, 6/1/2007
           14173720            12552200             4479031            16003017 D.J. Bernstein,2001 
       817201061520       2095956175810       2452045365504       2734283895746 Andrew Bremner, 2013
         4707813440          7813353720         11988496761         12558554489 Andrew Bremner, 2013
            1841160           121952168           122055375           145087793 Juergen Rathmann, 5/31/2007
      1041572957760        535914713672        187577183625       1059621884297 Seiji Tomita, 7/16/2017
           41084175            31669120             2164632            44310257 Robert Gerbicz, 11/08/2006
         1593513080          1553556440            92622401          1871713857 Robert Gerbicz, ...
           16281009             7028600             3642840            16430513 D.J. Bernstein,2001 
          769321280           606710871           558424440           873822121 Robert Gerbicz, ...
           65932985            42878560            10409096            68711097 Robert Gerbicz, 11/08/2006
          106161120            87865617            34918520           117112081 Robert Gerbicz, 11/02/2006

70 Digits (Elkies, 1988)

from ams.org:

9033964577482532388059482429398457291004947925005743028147465732645880^4 +
4417264698994538496943597489754952845854672497179047898864124209346920^4 + 
1439965710648954492268506771833175267850201426615300442218292336336633^4 =
9161781830035436847832452398267266038227002962257243662070370888722169^4