1.Introduction

About a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4.
Latest solution table of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 

Before, we showed some solutions about this equation.a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 
However, the resulting solutions were greater.
By the way, the first and the second smallest solution of this equation are below.

955^4 + 1770^4 + (-2634)^4 + 5400^4  = 5491^4 

48150^4 + (-31764)^4 + 27385^4 + 7590^4 = 51361^4 

This time, we found 3rd and 4th smallest solutions by Piezas[2]'s method.

1229559^4 + (-1022230)^4 + 1984340^4 + (-107110)^4 = 2084559^4

561760^4 + 1493309^4 + 3597130^4 + (-1953890)^4 = 3698309^4

My method depends on Piezas's method.


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)
     
According to Piezas[2], if simultaneous equation {(3),(4)} has a rational solution for some rational number m,
equation (1) has a rational solution.

So, we searched some m and found rational solutions of (1) when m=211/150.

By using a known soution (m,p,q)=( 211/150, 825,311), we can obtain a parametric solution of (3)
p= 75(2458031+737231k^2-670210k)/(112979+67021k^2)
q= (126281119+20843531k^2-50265750k)/(112979+67021k^2).......................(5)

Substitute (5) to (4),then we get a quartic equation. 

V^2 = 9471547265521339197853197k^4+22062657079770078500959500k^3+756411501601022852455206k^2
    + 5173869366750430780465500k-11494681792596210024563403..................(6)

Transform (6) to Weierstrass form (7).
    
Y^2 + XY = X^3 - X^2 + 1524020321143902735X - 752432065809125643921039075....(7)
   
We get a point P1(X,Y) = (6111852903633298501/4860181225, -19025135618414578117570247644/338827534100875)

As this point on the curve (7) does not have integer coordinates,
there are infinitely many rational points on the curve (7) by Nagell-Lutz theorem.

Furthermore, we get a point P2(X,Y) = (8983920690 , 859087441289655).
We can obtain some small solutions of (1) by using point P1 and P2.

Case 1: P1
The point corresponding to point P1 is Q1(48623/40973, -12031903478184804180000/1678786729).
Substitute k=48623/40973 to (5), we obtain a new solution of (1).


This is the 3rd smallest solution.

a =  1229559
b = -1022230
c =  1984340
d =  -107110
a+b+c+d = 2084559


Case 2: P2
The point corresponding to point P2 is Q2(39695363/6501037, 1270625227322688000/9409).
Substitute k=39695363/6501037 to (5), we obtain a new solution of (1).

This is the 4th smallest solution.

a =   561760
b =  1493309
c =  3597130
d = -1953890
a+b+c+d = 3698309


Case 3: 2P2
The point corresponding to point 2P2 is 2Q2(-8646082403/2246886247, 142738612065248148652750693812000/5048497806957745009).
Substitute k=-8646082403/2246886247 to (5), we obtain a new solution of (1).

a = -21781662390052941
b =  18242401493206790
c =  30883685043889070
d =   6072093865059140
a+b+c+d = 33416518012102059


Case 4: P1+P2
The point corresponding to point P1+P2 is Q(591502731203/883139537197, -159599624556047878563552000/173634831181249).
Substitute k=591502731203/883139537197 to (5), we obtain a new solution of (1).

a =  1619214280915810
b = -3065097088334401
c =  2940554547668260
d =  2243306697670930 
a+b+c+d = 3737978437920599



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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