1.Introduction

We show the parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^n, n=5,6,7,9 where Z=X+Y.
For n=2 and 3, please see [1],[2],and [3].


2.Method

X^4 + Y^4 + Z^4 + W^4 = T^n

Set X=a, Y=b, Z=a+b

Above equation becomes to 2(a^2+ab+b^2)^2+W^4=T^n.

Let (a,b,W,T)=(a0,b0,W0,T0) is a known solution.

a0^4 + b0^4 + (a0+b0)^4 + W0^4 = T0^n
Let (a,b) is a solution of a^4 + b^4 + (a+b)^4 + W0^4 = T0^n, then 2(a^2+ab+b^2)^2=T0^n-W0^4.

Hence (a,b) is parameterized by k as below.
a=Q(k)/P(k), b=R(k)/P(k), P(k)=1+k+k^2, Q(k),R(k) are polynomials of k.

Q(k)^4 + R(k)^4 + ( (Q(k)+R(k) )^4 + ((1+k+k^2)W0)^4 = (1+k+k^2)^4T0^n
Find a rational solution of 1+k+k^2 = u^(s/4), s=LCM(n,4), we get the general solution form as below.

Q(k)^4 + R(k)^4 + ( (Q(k)+R(k) )^4 + (u^(s/4)W0)^4 = (u^(s/n)T0)^n..................(1)


3.Results
 
n=5
According to 26^4+11^4+37^4+19^4=19^5, we get a parametric solution below.

First, we obtain a parametric solution a^2+a*b+b^2=1083 using (a0,b0)=(26,11), then we obtain
(X,Y,Z,W,T)=(-37-22*k+26*k^2, 11-52*k-37*k^2, -26-74*k-11*k^2, 19+19*k+19*k^2).
Hence X^4 + Y^4 + Z^4 + W^4=19^5*(1+k+k^2)^4.
Next, we find a rational solution of 1+k+k^2=u^5.
Set k=q/p. Let (p-q*w)=(m-n*w)^5 where w=(-1+sqrt(-3))/2 then we obtain (p,q) as follows.
Substitute (p,q)=(m^5-10*m^3*n^2+n^5-10*m^2*n^3,5*m^4*n-5*m*n^4+10*m^3*n^2-n^5) to equation (1),
then we obtain a parametric solution as follows.

X=-37*m^10-110*n*m^9+1170*n^2*m^8+4440*n^3*m^7+2310*n^4*m^6-6552*n^5*m^5-7770*n^6*m^4
 -1320*n^7*m^3+1170*n^8*m^2+370*n^9*m+11*n^10
 
Y=11*m^10-260*n*m^9-1665*n^2*m^8-1320*n^3*m^7+5460*n^4*m^6+9324*n^5*m^5+2310*n^6*m^4
 -3120*n^7*m^3-1665*n^8*m^2-110*n^9*m+26*n^10
 
Z=-26*m^10-370*n*m^9-495*n^2*m^8+3120*n^3*m^7+7770*n^4*m^6+2772*n^5*m^5-5460*n^6*m^4
 -4440*n^7*m^3-495*n^8*m^2+260*n^9*m+37*n^10
 
W=19*(m^2+n*m+n^2)^5

T=19*(m^2+n*m+n^2)^4

n=6
According to 324^4+18^4+342^4+441^4=63^6, we get a parametric solution below.

In the same way as the case of n=5, substitute (p,q)=(m^3-3*m*n^2-n^3, 3*m^2*n+3*m*n^2) to equation (1),
then we obtain a parametric solution as follows.

X=-342*m^6-108*n*m^5+4860*n^2*m^4+6840*n^3*m^3+270*n^4*m^2-1944*n^5*m-342*n^6
 
Y=18*m^6-1944*n*m^5-5130*n^2*m^4-360*n^3*m^3+4860*n^4*m^2+2052*n^5*m+18*n^6
 
Z=-54*(-2*m^2+2*n*m+3*n^2)*(m^2+6*n*m+2*n^2)*(-3*m^2-4*n*m+n^2)
 
W=441*(m^2+n*m+n^2)^3

T=63*(m^2+m*n+n^2)^2

n=7
According to 65052^4+29646^4+35406^4+75555^4=657^7, we get a parametric solution below.

In the same way as the case of n=5, substitute (p,q)=(m^7-21*m^5*n^2+21*m^2*n^5-35*m^4*n^3+7*m*n^6,
7*m^6*n-35*m^3*n^4+n^7+21*m^5*n^2-21*m^2*n^5) to equation (1),
then we obtain a parametric solution as follows.

X=35406*m^14-415044*n*m^13-5919732*n^2*m^12-12887784*n^3*m^11+29675646*n^4*m^10+130234104*n^5*m^9
 +106324218*n^6*m^8-101745072*n^7*m^7-195351156*n^8*m^6-70882812*n^9*m^5+29675646*n^10*m^4
 +23678928*n^11*m^3+3221946*n^12*m^2-415044*n^13*m-65052*n^14
 
Y=29646*m^14+910728*n*m^13+3221946*n^2*m^12-10791144*n^3*m^11-65117052*n^4*m^10-70882812*n^5*m^9
 +89026938*n^6*m^8+223258464*n^7*m^7+106324218*n^8*m^6-59351292*n^9*m^5-65117052*n^10*m^4
 -12887784*n^11*m^3+2697786*n^12*m^2+910728*n^13*m+35406*n^14
 
Z=65052*m^14+495684*n*m^13-2697786*n^2*m^12-23678928*n^3*m^11-35441406*n^4*m^10+59351292*n^5*m^9
 +195351156*n^6*m^8+121513392*n^7*m^7-89026938*n^8*m^6-130234104*n^9*m^5-35441406*n^10*m^4
 +10791144*n^11*m^3+5919732*n^12*m^2+495684*n^13*m-29646*n^14
 
W=75555*(m^2+n*m+n^2)^7

T=657*(m^2+n*m+n^2)^4

n=9
According to 647^4+601^4+46^4+361^4=19^9, we get a parametric solution below.

In the same way as the case of n=5, substitute (p,q)=(m^9-36*m^7*n^2+126*m^4*n^5-9*m*n^8-84*m^6*n^3+84*m^3*n^6-n^9,
9*m^8*n-126*m^5*n^4+36*m^2*n^7+36*m^7*n^2-126*m^4*n^5+9*m*n^8) to equation (1),
then we obtain a parametric solution as follows.

X=46*m^18-10818*n*m^17-98991*n^2*m^16-37536*n^3*m^15+1839060*n^4*m^14+5543496*n^5*m^13+853944*n^6*m^12
 -19126224*n^7*m^11-28311426*n^8*m^10-2236520*n^9*m^9+26298558*n^10*m^8+20590128*n^11*m^7+853944*n^12*m^6
 -5149368*n^13*m^5-1979820*n^14*m^4-37536*n^15*m^3+91953*n^16*m^2+11646*n^17*m+46*n^18
 
Y=601*m^18+11646*n*m^17+7038*n^2*m^16-490416*n^3*m^15-1979820*n^4*m^14-394128*n^5*m^13+11156964*n^6*m^12
 +20590128*n^7*m^11+2012868*n^8*m^10-29220620*n^9*m^9-28311426*n^10*m^8-1463904*n^11*m^7+11156964*n^12*m^6
 +5543496*n^13*m^5+140760*n^14*m^4-490416*n^15*m^3-98991*n^16*m^2-828*n^17*m+601*n^18
 
Z=647*m^18+828*n*m^17-91953*n^2*m^16-527952*n^3*m^15-140760*n^4*m^14+5149368*n^5*m^13+12010908*n^6*m^12
 +1463904*n^7*m^11-26298558*n^8*m^10-31457140*n^9*m^9-2012868*n^10*m^8+19126224*n^11*m^7+12010908*n^12*m^6
 +394128*n^13*m^5-1839060*n^14*m^4-527952*n^15*m^3-7038*n^16*m^2+10818*n^17*m+647*n^18
 
W=361*(m^2+n*m+n^2)^9

T=19*(m^2+n*m+n^2)^4



4.References
 
[1].Tomita Seiji: {X^4 + Y^4 + Z^4 + W^4 = T^2}
[2].Tomita Seiji: {X^4 + Y^4 + Z^4 + W^4 = T^2}
[3].Tomita Seiji: {X^4 + Y^4 + Z^4 + W^4 = T^3}