1.Introduction

We show that ax^5 + by^5 + cz^5 = w^2 has infinitely many integer solutions.



2.Theorem
        
     

    If ax0^5 + by0^5 + cz0^5 = w0^2 and a+b+c=0, then ax^5 + by^5 + cz^5 = w^2 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 = -(5*c*z0+5*b*y0-m^2+5*a*x0)*(10*c*z0^2+5*a*x0^2+10*b*y0^2-2*m*n-5*x0*c*z0-5*x0*b*y0+x0*m^2)
              
    y = -(5*c*z0+5*b*y0-m^2+5*a*x0)*(10*c*z0^2+10*a*x0^2+5*b*y0^2-2*m*n-5*y0*c*z0+y0*m^2-5*y0*a*x0)
    
    z = -(5*c*z0+5*b*y0-m^2+5*a*x0)*(5*c*z0^2+10*a*x0^2+10*b*y0^2-2*m*n-5*z0*b*y0+z0*m^2-5*z0*a*x0)
    
    w = (100*m*a^2*x0^4-50*n*c^2*z0^3-50*n*b^2*y0^3+25*w0*c^2*z0^2+2*m^3*n^2+w0*m^4+200*m*c*z0^2*a*x0^2
      +200*m*c*z0^2*b*y0^2+200*m*a*x0^2*b*y0^2-50*n*c*z0^2*b*y0-50*n*c*z0^2*a*x0-50*n*a*x0^2*c*z0
      -50*n*a*x0^2*b*y0-50*n*b*y0^2*c*z0-50*n*b*y0^2*a*x0+50*w0*c*z0*b*y0+50*w0*c*z0*a*x0+50*w0*b*y0*a*x0
      +100*m*c^2*z0^4+100*m*b^2*y0^4-50*n*a^2*x0^3+25*w0*b^2*y0^2+25*w0*a^2*x0^2-30*c*z0^2*m^2*n
      -30*a*x0^2*m^2*n-30*b*y0^2*m^2*n+10*m*n^2*c*z0+10*m*n^2*b*y0+10*m*n^2*a*x0-10*w0*c*z0*m^2
      -10*w0*b*y0*m^2-10*w0*m^2*a*x0)*(5*c*z0+5*b*y0-m^2+5*a*x0)^3
      
    where m = -5/8*(-8*a*x0^3*w0^2-8*c*z0^3*w0^2+5*a^2*x0^8+10*a*x0^4*b*y0^4+10*a*x0^4*c*z0^4+5*b^2*y0^8+10*b*y0^4*c*z0^4+5*c^2*z0^8-8*b*y0^3*w0^2)/(w0^3),
          n = 5/2*(a*x0^4+b*y0^4+c*z0^4)/w0.
            
 
Proof.

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

Let x=t+x0, y=t+y0, z = t+z0, w = mt^2+nt+w0..............................(2)

Substitute (2) to (1) and using a+b+c=0 and ax0^5 + by0^5 + cz0^5 = w0^2, and simplifying (1), we obtain

(5cz0+5by0-m^2+5ax0)t^4
+(10cz0^2+10ax0^2+10by0^2-2*mn)t^3
+(-2mw0+10ax0^3+10cz0^3-n^2+10by0^3)t^2
+(5cz0^4+5by0^4-2nw0+5ax0^4)t..............................................(3)

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

m = -5/8(-8ax0^3w0^2-8cz0^3w0^2+5a^2x0^8+10ax0^4by0^4+10ax0^4cz0^4+5b^2y0^8+10by0^4cz0^4+5c^2z0^8-8by0^3w0^2)/(w0^3)

n = 5/2(ax0^4+by0^4+cz0^4)/w0.

Finally, we obtain t as follows,

t = -(10cz0^2+10ax0^2+10by0^2-2mn)/(5cz0+5by0-m^2+5ax0).

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

Finally, this new rational solution leads to a new integer solution. 

Hence if ax0^5 + by0^5 + cz0^5 = w0^2 and a+b+c=0, then ax^5 + by^5 + cz^5 = w^2 has infinitely many integer solutions.
   
Q.E.D.　
 
　　                  
       
3.Example

Case1: (a,b,c)=(2,1,-3), (x0,y0,z0,w0)=(1,8,3,179), 2x^5 + y^5 - 3z^5  = w^2

x = -57420605162786359789966219134527

y = 92228688300843426842575380895520

z = -14663664173177849323525761983085

w = 73666338671294218755146268085788369165939337858897337242600008821730257139957419


Case2: (a,b,c)=(2,-1,-1), (x0,y0,z0,w0)=(16,10,18,328), 2x^5 - y^5 - z^5  = w^2

x = -4122388081173080266664128

y = -4438765953618809778820534

z = -4016928790357837095945326

w = 19694515585707737018854692789165469865694929156624800421744008


Case3: (a,b,c)=(3,2,-5), (x0,y0,z0,w0)=(4,11,9,173), 3x^5 + 2y^5 - 5z^5  = w^2

x = -9778537404991387297302483096326676

y = 528459932347037681264298747690732331

z = 374677512417886518818126967465858329

w = -213333668432228338839602418047583286943027777390218157739859809140552675602711434991136827