1.Introduction

We show the parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^3.
Similarly, we will be able to obtain the parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^n, n > 4.

2.Theorem
      
 
There are many parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^3.

     
Proof.

X^4+Y^4+Z^4+W^4 = T^3..................................................(1)

Set X=a, Y=b, Z=a+b....................................................(2)

2(a^2+ab+b^2)^2+W^4=T^3................................................(3)

If we find the rational solution of equation (3), we can obtain the parametric solution of (1).

Some solutions were found where |a,b|<100 and W<1000 below.
(a,b,W,T)=(18, 9, 99, 459),(27, -18, 99, 459),(27, -9, 99, 459),(51, 42, 135, 747),(62, 36, 115, 657),
          (68, -34, 17, 289),(75, 63, 51, 747),(78, 14, 115, 657),(92, -78, 115, 657),(92, -14, 115, 657),
          (93, -51, 135, 747),(93, -42, 135, 747),(93, 42, 51, 747),(98, -62, 115, 657),(98, -36, 115, 657).
          
Incidentally, (a,b,W,T)=(18, 9, 99, 459),(27, -18, 99, 459), and (27, -9, 99, 459) are a part of the same solution group as follows.

Integral solutions of a^2+ab+b^2=567 are (a,b)=( 18, -27),( -18, 27),( -9, 27),( 27, -18),( 9, 18),( -27, 9),(-27, 18),( 9, -27),
( -9, -18,),(18, 9),( 27, -9),( -18, -9).

I guess that equation (1) has infinitely many parametric solutions.

We show two examples for (a,b,W,T)=(18, 9, 99, 459) and (51, 42, 135, 747).

Q.E.D.


3.Examples
 

(a,b,W,T)=(18, 9, 99, 459)

(X,Y,Z,W,T)=(-27*m^6-54*n*m^5+270*n^2*m^4+540*n^3*m^3+135*n^4*m^2-108*n^5*m-27*n^6,
             9*m^6-108*n*m^5-405*n^2*m^4-180*n^3*m^3+270*n^4*m^2+162*n^5*m+9*n^6,
             -18*m^6-162*n*m^5-135*n^2*m^4+360*n^3*m^3+405*n^4*m^2+54*n^5*m-18*n^6,
             99*(m^2+n*m+n^2)^3,
             459*(n^2+n*m+m^2)^4)

(a,b,W,T)=(51, 42, 135, 747)

(X,Y,Z,W,T)=(-93*m^6-252*n*m^5+765*n^2*m^4+1860*n^3*m^3+630*n^4*m^2-306*n^5*m-93*n^6,
             42*m^6-306*n*m^5-1395*n^2*m^4-840*n^3*m^3+765*n^4*m^2+558*n^5*m+42*n^6,
             -51*m^6-558*n*m^5-630*n^2*m^4+1020*n^3*m^3+1395*n^4*m^2+252*n^5*m-51*n^6,
             135*(m^2+n*m+n^2)^3,
             747*(n^2+n*m+m^2)^4)


4.Reference
 
[1].Tito Piezas: {16. v^4+x^4+y^4+z^4=nt^k}