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

Allan J. MacLeod[1] showed some new solutions of Jacobi-Madden equation.
On this occcasion, we organaized all the known solutions. 
According to Piezas[2], if simultaneous equation {(3),(4)} has a rational solution for some rational number m,
equation (1) has a rational solution.

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

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

(m^2-7)p^2+24pq-24q^2=(m^2+1)r^2....................(3)

8mp^2-24mpq-3(m^2-8m+1)q^2=(m^2+1)s^2...............(4)

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

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

m = (c^2+cd+d^2)/((c+b+d)(a+c+d))...................(6)

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

3597130^4 + 1493309^4 + 561760^4 + (-1953890)^4 = 3698309^4
m=211/150, 361/61, 2041/150, 2191/1891 give above solution.

841263^4 + (-792940)^4 + (-44410)^4 + 3852350^4 = 3856263^4
m=157/150, 307/7, 8467/150, 8617/8317 give above solution.

Jeremey Rouse pointed out that m1=(m+1)/(m-1) is solvable if m is solvable.
For example, if m=211/150 then m1=(m+1)/(m-1)=361/61.
Using m=211/150 and m=361/61, we obtain same solution as follows.
(-107110)^4 + 1984340^4 + 1229559^4 + (-1022230)^4 = 2084559^4
When m=211/150, simultaneous equations {equation (3)=0,equation (4)=0} is transformed to elliptic curve 
Y^2 + XY = X^3 - X^2 + 1524020321143902735X - 752432065809125643921039075.
Similarly, when m=361/61 we obatain Y^2 + XY = X^3 - X^2 + 1524020321143902735X - 752432065809125643921039075.
In this way, when m=211/150 and m=361/61 we obtain same elliptic curve.





For details for each solutions, please see here.
a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 1 
a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 2 
a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 3 



   
   m                        a                    b                    c                      d                  a+b+c+d   Discoverer

              
2521/325                   5400                -2634                  1770                   955                   5491  Brudno 1964 
                
1807/475                  48150               -31764                 27385                  7590                  51361  Jaroslaw Wroblewski 
               
961/61               1338058950            -89913570             504106884            -404747255             1347505009  Jacobi-Madden 2008 
          
961/61               3095408880           1655829870            -157072326           -1406590625             3187575799  Jacobi-Madden 2008 
          
1807/475             4625798910             46140636            3744195265           -3172936050             5243198761  Jacobi 2010 
          
1651/126           115711769730         -64829623500           10424211666           58931380645           120237738541  Jacobi-Madden 2008 
        
961/61     65839853029653125119 -2344554049955763482  13576788496021617746  10170481573786789344   66141582599903897381  Jacobi-Madden 2008 
           989570               9062920               750526                081475                 758651 

961/61     23386662965142418709 14414475289412269061  -8205735876023550873  -5313492406384356182   24281909972146780715  Jacobi-Madden 2008 
           2091730              9290650                3782724               6924255               0675401

1651/126   28653793324729854441 -4244724647017458654  11559123698010568095  -7096774875862844712   28871417499860119170  Jacobi-Madden 2008 
           97526070              32372850              49486884              68805095              45835009

2521/325   13460314194003990473 -1090767881586925080  76168461612680198065  48509574010252242359  150204389404279837134  Jacobi-Madden 2008 
           258785230             2298477780           11367374              43996925              15671749 

1069/169              378573600            145514934              65167315            -201317790              387938059  Seiji Tomita 2009 
157/150                  841263              -792940                -44410               3852350                3856263  Jeremey Rouse 2014
157/150             -7784423350          -2943361793           10692790190          -49328431840           -49363426793  Seiji Tomita 2017
157/150            -35835675310         168853510327         -134075405440         -644955984250          -646013554673  Seiji Tomita 2017
31/6                   39913670            -23859495              15187700              10116014               41357889  Jeremey Rouse 2014
31/6                   53902630              2542025              35847220             -34122866               58169009  Jeremey Rouse 2014
31/6            -34998027446475      -32309023830920        42457132181770       -25125873749306        -49975792844931  Seiji Tomita 2017
31/6            -29175553438600       -5059968816155        21271610809130       -19158210038746        -32122121484371  Seiji Tomita 2017
1807/475          1058103081810         535945811334        -1140105961325          944080652640          1398023584459  Seiji Tomita 2009
211/150                 1229559             -1022230               1984340               -107110                2084559  Seiji Tomita 2015
211/150                  561760              1493309               3597130              -1953890                3698309  Seiji Tomita 2015
211/150      -21781662390052941    18242401493206790     30883685043889070      6072093865059140      33416518012102059  Seiji Tomita 2015
211/150        1619214280915810    -3065097088334401      2940554547668260      2243306697670930       3737978437920599  Seiji Tomita 2015
3523/2623             336869940           -178944510            -210240721            -396470430             -448785721  Seiji Tomita 2015
2977/2502             719130355          -2889516060            4672341330            2405612802             4907568427  Seiji Tomita 2015
2851/1626               2434795             -1945570              -1483582              -1858600               -2852957  Seiji Tomita 2015
1159/259              753684930            294589950             558360120            -701876813              904758187  Seiji Tomita 2015
1159/259              500764020           1768211850            1297734853           -1510410870             2056299853  Seiji Tomita 2015
331/31                732896170            303742360             189854902            -460945405              765548027  Seiji Tomita 2015
331/31          530920858665230      377970149282480        35966749745415      -360346958398438        584510799294687  Seiji Tomita 2015
331/31            -150723250810        1751113229630          802797814305         -626137906588          1777049886537  Seiji Tomita 2015
217/25          237321095011880     -558974521862416       -22424373335225      -222795507072280       -566873307258041  Seiji Tomita 2015
217/25            -259448373800       -1526478290216          889698809680         -687020381505         -1583248235841  Seiji Tomita 2015
373/150             50627178820           1357751663           55867457830          -41572821650            66279566663  Seiji Tomita 2015
373/150       -7929822455879583    10830318289720550      9309384955649330       392431543415120      12602312332905417  Seiji Tomita 2015
1807/475   75600593267607958148 25491945779594685472  64414567706058373954  -7184716582116907200   93659940932091945568  Seiji Tomita 2010 
           0132094              1610090               1816400                60678365              2880219 

1807/475   47889177339639353814 -6535571193228450354  -2648566219395408586  38071142693337417804   52939086645794235394  Seiji Tomita 2010 
           221283135714         843506097540           9452902545110         472545034525          397419527589

1807/475  -9920529841425116466 28621756052395468228  21790201280933421044  -9566517417742495552   30924910074161277255  Seiji Tomita 2009 
          28468071089452679223 99562799473665653544  86291565186506687683  44337690029230422403   13048603541489239601
          82794305394          01483969850           93853565040           04835226765            07708002731

499/474          3868630767650         895775733285        21271390911326       -4745425061560          21290372350701   Allan J. MacLeod 2017
511/150            -6714317914         994485789915         -698106854980         864417463190           1154082080211   Allan J. MacLeod 2017
3163/1350         -16515508578          10824551825          -15627586290           1711841340            -19606701703   Allan J. MacLeod 2017
18913/438         123140611690         446604426005          -96985017746         -25263498320            447496521629   Allan J. MacLeod 2017
1213/438          106185491830          80795489585          146163232960        -149806955726            183337258649   Allan J. MacLeod 2017
1963/150       662971279500154      309770790508565        85290604949260     -371936154165950         686096520792029   Allan J. MacLeod 2017


References

[1]:Allan J. MacLeod, Computational Experiments on a^4 + b^4 + c^4 + d^4=(a+b+c+d)^4,https://arxiv.org/pdf/1704.03200
[2]:Tito Piezas, More elliptic curves for a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 



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