1.Introduction

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

For more information, please see a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4. 

This time, new kinds of solutions were found by Piezas[2]'s method.

Added m=157/150 (2017.7.5).

(-7784423350)^4 + (-2943361793)^4 + 10692790190^4 + (-49328431840)^4 = (-49363426793)^4
(-35835675310)^4 + 168853510327^4 + (-134075405440)^4 + (-644955984250)^4 = (-646013554673)^4
 

Added m=31/6 (2017.7.3).

(-34998027446475)^4 + (-32309023830920)^4 + 42457132181770^4 + (-25125873749306)^4 = (-49975792844931)^4
(-29175553438600)^4 + (-5059968816155)^4 +  21271610809130^4 + (-19158210038746)^4 = (-32122121484371)^4  
 

Added m=2977/2502 and m=3523/2623(2015.8.27).

Added m=2851/1626(2015.8.25).( one small solution was found!)

2434795^4 + (-1945570)^4 + (-1483582)^4 + (-1858600)^4 = (-2852957)^4

Added m=1159/259(2015.8.24).

Added m=331/31(2015.8.13).

Added m=217/25 and m=373/150(2015.8.4).



2. New solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 

a^4 + b^4 + 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)


Case: (m,p,q)=( 3523/2623, -166/75, -520963/393450 )

V^2 = -331681663746513932364551k^4 -34688794836882579122981400k^3+ 56971035291780456815926734k^2
    -  92648436936497943453916200k+ 22076926949753911943770761

Y^2 + XY = X^3 - X^2 + 563409204410536798285763685X -2995335769290282031298386451938294362075

We get one point, P1(X,Y) = (3163565025826494197535058469190/34681504514224681, 5803474795055002524050552925526386043239167145/6458726493366248703247829).

The point corresponding to point P1 is Q1(124746/689545,1255704376111198922101263/475472307025).

By using k=124746/689545, we obtain a solution (1).


a =   336869940  
b =  -178944510
c =  -210240721
d =  -396470430
a+b+c+d = -448785721



Case: (m,p,q)=( 2977/2502  1693/575 270897/159850 )

V^2 = 2975464396988803385170821k^4+20299432132076564188222200k^3+47887584661836454812735966k^2
    + 73239977604483207257730600k+51867655959247737318942909

Y^2 + XY = X^3 - X^2 + 257601171907326617645457465X + 7793631714178739997725471735366587767741

We get one point, P1(X,Y) = (1424060202351065451671444952057/142723839434695744, -5746206542396291927618503550675789761042645383/53919366412052705987497472).

The point corresponding to point P1 is Q1(-44083/58497,4582746747059983326848/1140633003).

By using k=-44083/58497, we obtain a solution (1).


a =   719130355 
b = -2889516060
c =  4672341330
d =  2405612802
a+b+c+d = 4907568427



Case: (m,p,q)=( 2851/1626, -1873/385, -332677/313005 )

V^2 = 413311767680253945386291181k^4-276219739811225906909842860k^3-981803770684311390957676434k^2
    - 3957328185693703225615980k-495567955372658516362812531

Y^2 + XY = X^3 - X^2 + 971928215887051428490754955X -55329591449597546272216211300713415224779

We get one point, P1(X,Y) = (456381414199597952657263/79727041, 308317587090396226431520762632704024/711882749089).

The point corresponding to point P1 is Q1(-69781/48917,12090580828917642874368/2392872889).

By using k=-69781/48917, we obtain a solution (1).


a =  2434795
b = -1945570
c = -1483582
d = -1858600
a+b+c+d = -2852957



Case: (m,p,q)=( 1159/259, 109/30, -19889/15540)

V^2 = 5966583814727341728841k^4+1211722378372680362160k^3-14141049488981307041034k^2
    + 5549549340932544664080k-9839533155663041180271

Y^2 + XY = X^3 - X^2 + 20104584674898963456315X -2738788580074111460909980285884075

We get three points, P1(X,Y) = (226786433485296315053171420490/1588197127297246489, -110422368704764637199537350592153786563478585/2001504676704492494642378237) ,
P2(X,Y) = (94393240410, -47196620205) and P3(X,Y) = (154802565393, 63899184899972688).

The point corresponding to point P1 is Q1(165136/55357,1902748666611719820047/3064397449).

By using k=165136/55357, we obtain a solution (1).


a = 753684930
b = 294589950
c = 558360120
d =-701876813
a+b+c+d = 904758187


The point corresponding to point P2-P3 is Q2(-10862937467/2080989131, 12024169524425656949911252/6140998925581).

By using k=-10862937467/2080989131, we obtain a solution (1).


a =   500764020
b =  1768211850
c =  1297734853
d = -1510410870
a+b+c+d = 2056299853



Case: (m,p,q)=( 331/31, 187/180, -71/3720)

V^2 = 982036844785066121062510489k^4+1625169610515932429077896480k^3-14064673954444405979442135546k^2+
      21076926571614198303788588640k-9685611808219176923287213839

Y^2 + XY = X^3 - X^2 + 137120611612540455X + 26687311291242150339761325

We get three points, P1(X,Y) = (-163027230, 81513615) ,  P2(X,Y) = (-133444767, 2452201454148) and P3(X,Y) = (7167472770, 607632557013615).

The point corresponding to point P1 is Q1(97957085/85593233, -148644894042034347548516/12784598761).

By using k=97957085/85593233, we obtain a solution (1).


a = 732896170
b = 303742360
c = 189854902
d =-460945405
a+b+c+d = 765548027


The point corresponding to point P2 is Q2(2821887261/2216625509, -15558117257164118556707332788/922017010161289).

By using k=2821887261/2216625509, we obtain a solution (1).


a = 530920858665230
b = 377970149282480
c =  35966749745415
d =-360346958398438
a+b+c+d = 584510799294687


The point corresponding to point P3 is Q3(-684131628/39868877, 2538336661437577822783128417/298278730201).

By using k=-684131628/39868877, we obtain a solution (1).


a = -150723250810
b = 1751113229630
c =  802797814305
d = -626137906588
a+b+c+d = 1777049886537



Case: (m,p,q)=( 217/25, 50, 113)

V^2 = -21518624106078701615228k^4 + 1026953921106759839256k^2
    + 3160909881926921655972.................................................(5)

Transform (5) to Weierstrass form (6).

Y^2 + XY = X^3 - X^2 + 12999888419190300X + 61667321831119991097936..........(6)
   
We get two points, P1(X,Y) = (-4735512, 2367756) and P2(X,Y) = (5436363, 364002914631).

The point corresponding to point P1 is Q1(2709423/4293973, 232320351085655167161000/18438204124729).

By using k=-2709423/4293973, we obtain a solution (1).


a =  237321095011880
b = -558974521862416
c =  -22424373335225 
d = -222795507072280
a+b+c+d = -566873307258041


The point corresponding to point P2 is Q2(-9811767/20421592, -17383329082814625/366368).

By using k=-9811767/20421592, we obtain a solution (1).


a =  -259448373800
b = -1526478290216
c =   889698809680
d =  -687020381505
a+b+c+d = -1583248235841


Case: (m,p,q)=( 373/150, 1775, 549)


V^2 = 58230169713988077k^4+ 262109142307798500k^3 -384285292695045354k^2
    + 152187282413776500k -121423427787237723

Y^2 + XY = X^3 - X^2 + 32758984559287641735X - 958560858196498508253707949075

We get two points, P1(X,Y) = (8757606690, -4378803345) and  P2(X,Y) = (26157137313, 4218390024028644).

The point corresponding to point P1 is Q1(26091733/16015217, -206816371488340805808000/256487175557089).

By using k=26091733/16015217, we obtain a solution (1).


a = 50627178820
b =  1357751663
c = 55867457830
d =-41572821650
a+b+c+d = 66279566663


The point corresponding to point P2 is Q2(-2540333829971/313431118429, -5607756066292610164695264000/607805938288635229).

By using k=-2540333829971/313431118429, we obtain a solution (1).


a = -7929822455879583
b = 10830318289720550
c =  9309384955649330
d =   392431543415120
a+b+c+d = 12602312332905417


3.References

[1]Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236.

[2]Tito Piezas, More elliptic curves for a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 





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