Solution table of a^4 + b^4 + c^4 = d^4

Some new solutions of above equation were found recently.
On this occcasion, we organaized all the known solutions. 
According to Piezas[1], if simultaneous equation {(3),(4)} has a rational solution for some rational number m,
equation (1) has a rational solution.

a^4 + b^4 + c^4 = d^4.................................(1)

a=p+r, b=p-r, c=s, d=q................................(2)

-(3m^2-8m+6)p^2+2(m^2-2)pq-2mq^2=(m^2+2)r^2...........(3)

-4(m^2-2)p^2+8mpq+(m^2-2)q^2=(m^2+2)s^2...............(4)

m =(4p^2-q^2-s^2)/(3p^2-2pq+r^2)......................(5)

From equation (1),(2),(5), m is given below.

m = (a^2+2ab+b^2-d^2-c^2)/(a^2+ab+b^2-da-db)..........(6)

We used the equation (6) to obtain m from a known solution {a,b,c,d}.

For details for old results, please see here.
 Table of the solutions for A4+ B4+ C4= D4
 Table of the solutions for A4+ B4+ C4= D4 Part 2

Leonid Durman's project page. 
 Euler Quartic Conjecture.


New solutions were found.

(7/20/2017)
(m,a,b,c,d)=(-125/92, 17111129720, 52289667920, 55479193841, 64244765937)
(m,a,b,c,d)=(-125/92, 18776929334105, 23449050222680, 12035933588696, 25866132798297)

(7/16/2017)
(m,a,b,c,d)=(-5/8, 275156240, 219076465, 630662624, 638523249)
(m,a,b,c,d)=(-361/540, 1041572957760, 535914713672, 187577183625, 1059621884297)
(m,a,b,c,d)=(-41/36, 41714673255, 18125123544, 34169217200, 46055390617)

(7/15/2017)
(m,a,b,c,d)=(-29/12, 894416022327, 125777308440, 2032977944240, 2051764828361)

Search results.

m={-5/8, -477/692, -1617/200} sometimes give the same solutions.
m={-9/20, 1000/47, -1041/320, -1425/412}      "
m={-29/12, 1865/132}                          "
m={-93/80, -400/37}                           "
m={-136/133, 201/4, -1005/568}                " 
m={-361/540, 1861/240}                        "
m={-817/660, -1581/1520}                      " 



   m                       a                    b                    c                     d         Discoverer

-5/8                18796760              15365639               2682440              20615673  (Noam Elkies, 1986) 
-5/8                  414560                217519                 95800                422481  (Roger Frye, 1988)
-5/8       16921802132217020     15075240668820384    128805698258642759     16774794902382238  (Seiji Tomita, 03/13/2006)
                  4480680305           72584786800            1062203384           23661446513
-5/8               582665296             260052385             186668000             589845921  (Seiji Tomita, 03/13/2006)
-5/8              3134081336            2448718655             664793200            3393603777  (Seiji Tomita, 01/28/2007)
-5/8       47886740272114976      8813425670440240     56827813308111785     62940516903410601  (Seiji Tomita, 8/13/2008)
-5/8        3579087147375440     14890026433468471     18565945114216720     20249506709579721  (Seiji Tomita, 8/13/2008)
-5/8           4987588419655         2480452675600          502038853976         5062297699257  (Seiji Tomita, 5/15/2008)
-5/8               275156240             219076465             630662624            638523249   (Seiji Tomita, 7/16/2017)
-9/20                2767624               1390400                673865               2813001  (Allan MacLeod,1997) 
-9/20             1670617271             632671960              50237800            1679142729  (Seiji Tomita, 03/13/2006)
-9/20         22495595284040         7592431981391        27239791692640        29999857938609  (Seiji Tomita, 03/13/2006)
-9/20             5821981400           15355831360             140976551           15434547801  (Seiji Tomita, 10/24/2007)
-9/20        130064300991400       440804942580160       514818101299289       573646321871961  (Seiji Tomita, 9/15/2008)
-29/12               8332208               5507880               1705575               8707481  (D.J. Bernstein,2001) 
-29/12          894416022327          125777308440         2032977944240         2051764828361  (Seiji Tomita, 7/15/2017)
-41/36            1166705840             859396455             588903336            1259768473  (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov , 01/25/2008)
-41/36           41714673255           18125123544           34169217200           46055390617  (Seiji Tomita, 7/16/2017)
-5/44    1171431921806295280   5113654767499398206   4048310673060768880   5557992180974240706  (Andrew Bremner, 2013)
-93/80              11289040               8282543               5870000              12197457  (D.J. Bernstein,2001) 
-93/80            1662997663            1237796960             686398000            1787882337  (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov , 11/2/2007)
-125/92    54772209880944552     50049879917257109     16178554328069755     62586521087452988  (Andrew Bremner, 2013)
                   338211440             403510290             572637088             953161234
-125/92          17111129720           52289667920           55479193841           64244765937  (Seiji Tomita, 7/20/2017)
-125/92       18776929334105        23449050222680        12035933588696        25866132798297  (Seiji Tomita, 7/20/2017)
-136/133            27450160             146627384             108644015             156646737  (Juergen Rathmann, 6/1/2007)
-136/133            14173720              12552200               4479031              16003017  (D.J. Bernstein,2001) 
-56/165         817201061520         2095956175810         2452045365504         2734283895746  (Andrew Bremner, 2013)
233/60            4707813440            7813353720           11988496761           12558554489  (Andrew Bremner, 2013)
-361/540             1841160             121952168             122055375             145087793  (Juergen Rathmann, 5/31/2007)
-361/540       1041572957760          535914713672          187577183625         1059621884297  (Seiji Tomita, 7/16/2017)
-817/660            41084175              31669120               2164632              44310257  (Robert Gerbicz, 11/08/2006)
-865/592          1593513080            1553556440              92622401            1871713857  (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov , 10/31/2007)
12185/432           16281009               7028600               3642840              16430513  (D.J. Bernstein,2001) 
-12285/4112        769321280             606710871             558424440             873822121  (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov , 11/2/2007)
-21021/9788         65932985              42878560              10409096              68711097  (Robert Gerbicz, 11/08/2006)
25607745/5442892   106161120              87865617              34918520             117112081  (Robert Gerbicz, 11/02/2006)

Reference

[1]:Tito Piezas, More elliptic curves for a^4 + b^4 + c^4 = d^4 


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