1.Introduction

We showed a new solution of ax^4 + by^4 + cz^4 + dw^4 = 0 by using a known solution before.
This time, by supposing a+b+c+d= 0 without using a known solution, we show a new solution of ax^4 + by^4 + cz^4 + dw^4 = 0.


2.Theorem
        
     

    ax^4 + by^4 + cz^4 + dw^4 = 0 has a new solution as follows.
    By using this new solution as a known solution, we obtain infinitely many integer solutions.
    
    x = (4*c^2-4*b^2+4*a*c-4*a*b)*n-3*c^3*b+c^3*a-9*c^2*a*b-6*c*b*a^2+c^2*a^2-6*c^2*b^2-3*c*b^3+a^2*b^2-9*a*c*b^2+a*b^3
    
    y = (-4*c^2-4*b*c-8*a*c)*n-3*c^3*a+c^3*b+3*c^2*a*b+c*b^3+2*c^2*b^2-3*c^2*a^2+6*c*b*a^2+a^2*b^2+7*a*c*b^2+a*b^3
    
    z = (4*b*c+4*b^2+8*a*b)*n+c^3*a+c^3*b+7*c^2*a*b+2*c^2*b^2+c^2*a^2+6*c*b*a^2+3*a*c*b^2+c*b^3-3*a*b^3-3*a^2*b^2
    
    w = (4*a*c-4*a*b)*n+c^3*a+c^3*b+2*c^2*b^2+c^2*a^2-c^2*a*b+a*b^3-6*c*b*a^2+c*b^3-a*c*b^2+a^2*b^2
    
    condition: a+b+c+d= 0
               a*b*c*d=n^2

 
Proof.

ax^4 + by^4 + cz^4 + dw^4 = 0........................................(1)
Put x = pt+1, y = qt+1, z = rt+1, w = t+1............................(2)

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

(bq^4+ap^4+cr^4+d)t^4
+(4cr^3+4bq^3+4d+4ap^3)t^3
+(6d+6cr^2+6bq^2+6ap^2)t^2
+(4d+4ap+4cr+4bq)t=0.................................................(3)

Equating to zero the coefficient of t, then we obtain

r = -(d+ap+bq)/c.
By the assumption, d = -a-b-c.
Equating to zero the coefficient of t^2, then we obtain

q = 1/2(2bc+2ab+2b^2-2bap
  +2sqrt(2bca^2p-bca^2p^2+2bc^2ap+2b^2cap-bc^2ap^2
  -b^2ap^2c-bc^2a-bca^2-b^2ca))/(bc+b^2)   ..........................(4)

2a^2bcp-a^2bp^2c+2bc^2ap+2b^2cap-bc^2ap^2
-b^2ap^2c-a^2bc-ab^2c-bc^2a..........................................(5)    
    
Then, (5) must be square.
Simplifying (5), then (5) becomes to -abc(-1+p)^2(a+b+c).
Hence, to obtain the rational solition q of (4), abcd must be square.    
Put n^2= abcd,
q = 1/2(2ab+2bc-2bap+2b^2+2n(-1+p))/(bc+b^2)    
t = - coefficient of t^3 / coefficient of t^4.(omitted since result is tedious)

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

Thus, we can find a solution by supposing {a+b+c+d= 0,abcd=n^2}.

By using this new solution as a known solution, we can obtain infinitely many integer solutions.

   
Q.E.D.　
 
　　                  
       
3.Example


Case1: (a,b,c,d)=(8, -1, 2, -9), (x,y,z,w)=(37, 47, 23, 33), 8x^4-y^4+2z^4-9w^4 = 0
       By using this new solution as a known solution, we obtain other new solution below.
       (x,y,z,w)=(2536422091855129, 64067278329889, 3586470331669969, 2928574368531009)

Case2: (a,b,c,d)=(10, -3, 8, -15), (x,y,z,w)=(25, 31, 11, 17), 10x^4-3y^4+8z^4-15w^4 = 0

Case3: (a,b,c,d)=(16, -27, 12, -1), (x,y,z,w)=(13, 57, 97, 167), 16x^4-27y^4+12z^4-w^4 = 0

Case4: (a,b,c,d)=(16, -5, 9, -20), (x,y,z,w)=(61, 71, 39, 49), 16x^4-5y^4+9z^4-20w^4 = 0

Case5: (a,b,c,d)=(25, -6, 8, -27), (x,y,z,w)=(217, 239, 179, 201), 25x^4-6y^4+8z^4-27w^4 = 0