1.Introduction

We show that ax^4 + by^4 + cz^4 + dw^4 + eu^4= 0 has always a integer solution by supposing {a+b+c+d+e= 0, d=-c}.


2.Theorem
        
     

    ax^4 + by^4 + cz^4 + dw^4 + eu^4 = 0 has always a integer solution as follows.
    
    x = -(-22b^3a^2+13b^4a-c^2a^3+18b^2a^3+a^5-7ba^4+25bc^2a^2+3b^3c^2-3b^5-35b^2c^2a)c
    
    y = (-b^5+7b^4a+35bc^2a^2-25b^2c^2a+3a^5+22b^2a^3-3c^2a^3-18b^3a^2+b^3c^2-13ba^4)c
    
    z = b^6+a^6-8a^2bc^3-8ab^2c^3+16ab^3c^2-30a^2b^2c^2+16ba^3c^2+15a^4b^2+15a^2b^4-6a^5b-6ab^5-a^4c^2-20a^3b^3-b^4c^2
    
    w = 20a^3b^3-b^6-15a^2b^4+6ab^5-15a^4b^2+b^4c^2+a^4c^2-a^6-16ab^3c^2-16ba^3c^2+30a^2b^2c^2-8ab^2c^3-8a^2bc^3+6a^5b
    
    u = (a+b)(a^4-4a^3b+6a^2b^2-a^2c^2+6abc^2-4ab^3+b^4-b^2c^2)c
    
    condition: {a+b+c+d+e= 0, d=-c}.
    a,b,c are arbitrary.           

 
Proof.

ax^4 + by^4 + cz^4 + dw^4 + eu^4 = 0....................................(1)

Put x = pt+1, y = qt+1, z = rt+1, w = st+1, u=t+1.......................(2)

Let a+b+c+d+e= 0.

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

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

By the assumption, e=-a-b-c-d.

Equating to zero the coefficient of t, then we obtain s = -(-a-b-c-d+ap+bq+cr)/d.

Equating to zero the coefficient of t^2, then we obtain

r = 1/2(-2cap+2bc+2ac+2c^2+2cd-2cbq
  +2sqrt(-ca^2p^2d+2ca^2pd+2c^2apd-b^2cq^2d+2b^2cdq-2bcad
  -c^2ap^2d-c^2bq^2d+2c^2dbq-cd^2ap^2-cd^2bq^2+2cd^2ap
  +2cd^2bq-b^2cd-bc^2d-a^2cd-ac^2d-cd^2a-cd^2b-2capdbq
  +2capbd+2acdbq))/(cd+c^2)  ...........................................(4)

(-cd^2a-ac^2d-a^2cd)p^2+(2cd^2a-2acdbq+2ac^2d+2a^2cd+2bcad)p
+(-b^2cd-cd^2b-bc^2d)q^2+(2cd^2b+2b^2cd+2bc^2d+2bcad)q
-b^2cd-bc^2d-a^2cd-2bcad-cd^2b-cd^2a-ac^2d..............................(5)    
    
Then, equation (5) must be square.

Let q=-p+2 and d=-c, then equation (5) becomes to c^2(-1+p)^2(-b+a)^2.

t = - coefficient of t^3 / coefficient of t^4.(omitted since result is tedious)

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

Thus, we can obtain a integer solution by supposing {a+b+c+d+e= 0, d=-c}.

Similarly, we can obtain a integer solution when q=2p-1.
   
Q.E.D.　
 
　　                  
       
3.Examples



Case1: (a,b,c,d,e)=(1, 2, 2, -2, -3), (x,y,z,w,u)=(182, 66, 97, 159, 58),  x^4 + 2y^4 + 2z^4 - 2w^4 - 3u^4 = 0

Case2: (a,b,c,d,e)=(1, 2, 3, -3, -3), (x,y,z,w,u)=(57, 21, 34, 47, 18),  x^4 + 2y^4 + 3z^4 - 3w^4 - 3u^4 = 0

Case3: (a,b,c,d,e)=(10, 11, 12, -12, -21), (x,y,z,w,u)=(1862652, 1033836, 2885773, 2920307, 1448244),  10x^4 + 11y^4 + 12z^4 - 12w^4 - 21u^4 = 0