1.Introduction

Moody and  Zargar[1] showed that x^4 + y^4 -2z^4 -2w^4 = 0 has infinitely many rational solutions.
It seems that Richmond showed if a rational solution to ax^4 + by^4 + cz^4 + dw^4 = 0 is known then others can be found.
Anyway, we show a new solution of ax^4 + by^4 + cz^4 + dw^4 = 0 by using a known solution.
By repeating this process, we can obtain infinitely many integer solutions.

2.Theorem
        
     

    ax^4 + by^4 + cz^4 + dw^4 = 0 has a new solution as follows.
    By repeating that we use a new solution as a known solution, we obtain infinitely many integer solutions.

    x = x0*{(y0^4*b*x0^12+c*x0^12*z0^4)*a^3+(2*y0^8*b^2*x0^8+7*y0^4*c*b*x0^8*z0^4+c^2*z0^8*x0^8)*a^2
          +(y0^12*b^3*x0^4+3*y0^8*c*b^2*x0^4*z0^4+(-4*y0^6*n*x0^6*z0^2*w0^2+6*y0^4*c^2*z0^8*x0^4)*b)*a
          -3*y0^12*c*b^3*z0^4+(-3*y0^8*c^2*z0^8-4*n*x0^2*z0^2*w0^2*y0^10)*b^2-8*y0^6*c*b*n*x0^2*z0^6*w0^2}
   
    y = y0*{(-3*c*x0^12*z0^4+y0^4*b*x0^12)*a^3+(2*y0^8*b^2*x0^8+3*y0^4*c*b*x0^8*z0^4-3*c^2*z0^8*x0^8+4*y0^2*z0^2*x0^10*w0^2*n)*a^2
          +(y0^12*b^3*x0^4+7*y0^8*c*b^2*x0^4*z0^4+(4*y0^6*n*x0^6*z0^2*w0^2+6*y0^4*c^2*z0^8*x0^4)*b+8*y0^2*c*z0^6*x0^6*w0^2*n)*a
          +y0^12*c*b^3*z0^4+y0^8*c^2*b^2*z0^8}
          
    z = z0*{(3*y0^4*b*x0^12-c*x0^12*z0^4)*a^3+(6*y0^8*b^2*x0^8+4*y0^2*z0^2*x0^10*w0^2*n+9*y0^4*c*b*x0^8*z0^4-c^2*z0^8*x0^8)*a^2
          +(3*y0^12*b^3*x0^4+9*y0^8*c*b^2*x0^4*z0^4+6*y0^4*c^2*b*z0^8*x0^4+4*y0^2*c*z0^6*x0^6*w0^2*n)*a
          -y0^12*c*b^3*z0^4+(-y0^8*c^2*z0^8-4*n*x0^2*z0^2*w0^2*y0^10)*b^2-4*y0^6*c*b*n*x0^2*z0^6*w0^2}
          
    w = w0*{(y0^4*b*x0^12+c*x0^12*z0^4)*a^3+(2*y0^8*b^2*x0^8-y0^4*c*b*x0^8*z0^4+c^2*z0^8*x0^8)*a^2
          +(-y0^8*c*b^2*x0^4*z0^4+y0^12*b^3*x0^4-6*y0^4*c^2*b*z0^8*x0^4-4*y0^2*c*z0^6*x0^6*w0^2*n)*a
          +y0^8*c^2*b^2*z0^8+y0^12*c*b^3*z0^4+4*y0^6*c*b*n*x0^2*z0^6*w0^2}
    
    condition: ax0^4 + by0^4 + cz0^4 + dw0^4 = 0
               a*b*c*d=n^2

 
Proof.

ax^4 + by^4 + cz^4 + dw^4 = 0.................................................(1)
Let {x0,y0,z0,w0} is a known solution.
Put x = pt+x0, y = qt+y0, z = rt+z0, w =st+w0.................................(2)

Substitute (2) to (1), and simplifying (1), we obtain

(bq^4+ap^4+cr^4+ds^4)t^4
+(4cz0r^3+4by0q^3+4dw0s^3+4ax0p^3)t^3
+(6dw0^2s^2+6cz0^2r^2+6by0^2q^2+6ax0^2p^2)t^2
+(4dw0^3s+4ax0^3p+4cz0^3r+4by0^3q)t............................................(3)
Equating to zero the coefficient of t and t^2, then we obtain

s = -(ax0^3p+cz0^3r+by0^3q)/(dw0^3)
r = 1/2(-2z0^2ax0^3pc-2z0^2cby0^3q
    +2sqrt(-c^2z0^4by0^2q^2dw0^4-c^2z0^4ax0^2p^2dw0^4-cdw0^4a^2x0^6p^2-cdw0^4b^2y0^6q^2-2cdw0^4ax0^3pby0^3q-cd^2w0^8by0^2q^2-cd^2w0^8ax0^2p^2))
    /((c^2z0^4+cdw0^4)z0)......................................................(4)
    
(-cdw0^4a^2x0^6-c^2z0^4ax0^2dw0^4-cd^2w0^8ax0^2)p^2
-2cdw0^4ax0^3pby0^3q
+(-c^2z0^4by0^2dw0^4-cd^2w0^8by0^2-cdw0^4b^2y0^6)q^2...........................(5)    
Then, (5) must be square.
Simplifying (5), (5) becomes to abcdx0^2y0^2w0^4(y0p-x0q)^2.
Hence, to obtain the rational solition r of (4), abcd must be square.    
Put n^2= abcd,
r = 1/2(-2z0^2ax0^3pc-2z0^2cby0^3q+2nx0y0w0^2(y0p-x0q))/((c^2z0^4+cdw0^4)z0)   
    
t = - coefficient of t^3 / coefficient of t^4.(omitted since result is tedious)

Substitute s, r, and t to (2), and remove common factors, then obtain a new solution.

Thus, we can find other solution by using known solution {x0,y0,z0,w0}.

By repeating this process, we can obtain infinitely many integer solutions.

   
Q.E.D.　
 
　　                  
       
3.Example


Case: (a,b,c,d)=(1,1,-2,-2), (x0,y0,z0,w0)=(19, 21, 7, 20), x^4 + y^4 -2z^4 -2w^4 = 0

x = 1086629
y = 3956211
z = 2724463
w = 2872540

We obtain same solution as Moody's one.

Case: (a,b,c,d)=(1,1,-17,-17), (x0,y0,z0,w0)=(13, 8, 5, 6), x^4 + y^4 -17z^4 -17w^4 = 0

x = 95896333
y = 176117272
z = 88538885
w = 18418554


Case: (a,b,c,d)=(1,1,-41,-41), (x0,y0,z0,w0)=(33, 17, 7, 13), x^4 + y^4 -41z^4 -41w^4 = 0

x = 220547349
y = 1190824381
z = 470722651
w = 50451089


Case: (a,b,c,d)=(6,3,-2,-1), (x0,y0,z0,w0)=(30, 9, 1, 47), 6x^4 + 3y^4 -2z^4 -w^4 = 0

x = 6494190
y = 2027781
z = 658291
w = 10175923


 




4.Reference


[1]. Dustin Moody and Arman Shamsi Zargar, On integer solutions of x^4+y^4-2z^4-2w^4=0,Notes on Number Theory and Discrete Mathematics
Vol.19, 2013, No.1