1.Introduction


We showed a new solution of ax^5 + by^5 + cz^5 = w^2 by using a known solution before.
This time, without using a known solution, we show a new solution of ax^5 + by^5 + cz^5 = w^2.


2.Theorem
        
     

    ax^5 + by^5 + cz^5 = w^2 has a new solution as follows.
    By using this new solution as a known solution, we obtain infinitely many integer solutions.

    x = (2000*a^4-4000*a^3*p*u^2+1400*a^2*p^2*u^4+600*a*p^3*u^6-19*p^4*u^8)
        *p*(10*a-9*p*u^2)*(200*a^3-60*a^2*p*u^2-154*a*p^2*u^4+3*p^3*u^6)
        
    y = -(2000*a^4-4000*a^3*p*u^2+1400*a^2*p^2*u^4+600*a*p^3*u^6-19*p^4*u^8)
        *p*(10*a-p*u^2)*(200*a^3-540*a^2*p*u^2+326*a*p^2*u^4+11*p^3*u^6)    
        
    z = 8*(2000*a^4-4000*a^3*p*u^2+1400*a^2*p^2*u^4+600*a*p^3*u^6-19*p^4*u^8)
        *(2*a-p*u^2)*(100*a^2-100*a*p*u^2+p^2*u^4)*p^2*u^2
            
    w = -p^3*u*(60*a^2-60*a*p*u^2-p^2*u^4)*(200000*a^6-600000*a^5*p*u^2+718000*a^4*p^2*u^4-436000*a^3*p^3*u^6+119020*a^2*p^4*u^8-1020*a*p^5*u^10+101*p^6*u^12)
        *(2000*a^4-4000*a^3*p*u^2+1400*a^2*p^2*u^4+600*a*p^3*u^6-19*p^4*u^8)^3
            
    condition: b = a-pu^2, c = -2a+pu^2
    a, p, u  : arbitrary
 
Proof.

ax^5 + by^5 + cz^5 = w^2..................................................(1)

Let x=t+p, y=t-p, z = t, w = mt^2+nt+s....................................(2)

Substitute (2) to (1) and using a+b+c=0 and (a-b)p^5=s^2, and simplifying (1), we obtain

(-m^2-5bp+5ap)t^4
+(-2mn+10ap^2+10bp^2)t^3
+(-2ms-n^2+10ap^3-10bp^3)t^2
+(5bp^4-2ns+5ap^4)t.......................................................(3)

Let a-b=pu^2, then we obtain b=a-pu^2 and s=p^3u.

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

m = -5/8(20a^2-20apu^2-3p^2u^4)/(pu^3)

n = 5/2p(2a-pu^2)/u.

Finally, we obtain t as follows

t = 8(-200a^3+300a^2pu^2-102ap^2u^4+p^3u^6)p^2u^2
/(-2000a^4+4000a^3pu^2-1400a^2p^2u^4-600ap^3u^6+19p^4u^8).

Substitute m, n, and t to (2), and obtain a solution.            


   
Q.E.D.　
 
　　                  
       
3.Example


Case1: (a,b,c)=(2,1,-3), 2x^5 + y^5 - 3z^5  = w^2

x = 78693505

y = -13270417

z = 32711544

w = 76960250193089506439

Case2: (a,b,c)=(2,-1,-1), 2x^5 - y^5 - z^5  = w^2

x = 656453091

y = -1131199635

z = -237373272

w = 45790804345280996227347

Case3: (a,b,c)=(3,2,-5), 3x^5 + 2y^5 - 5z^5  = w^2

x = 6319840401

y = -3032081921

z = 1643879240

w = -5447229082903918199130599