1.Introduction

A. Bremner and M. Ulas([1]) gave 2 families of solutions of w^2 = x^6 + y^6 + z^6.
The last time, we showed new 4 families of solutions of w^2 = x^6 + y^6 + z^6 by Bremner and Ulas's method.
This time, we add new 3 families of solutions of w^2 = x^6 + y^6 + z^6.


2.Method
        

w^2 = x^6 + y^6 + z^6.................................................................................(1)

Set w = z^3 + u(x^2+y^2) ,u(w+z^3) = x^4-x^2y^2+y^4, then we obtain

x^4-x^2y^2+y^4-u^2(x^2+y^2)-2uz^3.....................................................................(2)

We consider the following identity.

x^4-x^2*y^2+y^4-u^2*(x^2+y^2)-2*u*z^3

=(c*u*x-c*u*z+c*x^2-c*x*z-(b+c)*y^2-b*z^2)*((-y^2*b+b*u^2+b*x^2+b*z^2-y^2*c-c*u*x+c*u*z+c*u^2+c*x*z)/((b+c)^2))
+((x^2+u*x-x*z-u*z+z^2))*((c^2+b*c+b^2)*x^2+((-b^2-2*b*c)*u+(2*b*c+b^2)*z)*x+(-b*c-c^2)*u^2+(-2*b*c-2*c^2-b^2)*z*u+b^2*z^2)/(b+c)^2.

c*u*x-c*u*z+c*x^2-c*x*z-(b+c)*y^2-b*z^2=0.............................................................(3)

(c^2+b*c+b^2)*x^2+((-b^2-2*b*c)*u+(2*b*c+b^2)*z)*x+(-b*c-c^2)*u^2+(-2*b*c-2*c^2-b^2)*z*u+b^2*z^2=0....(4)

Hence if simultaneous equation {equation (3), equation (4)} has infinitely many rational solutions,
equation (1) has infinitely many integer solutions.

     
3.Results


Case 1: 2x^2+(2u-2z)x-2zu-3z^2-5y^2=0.................................(5)
       19x^2+(-21u+21z)x-10u^2-29zu+9z^2=0............................(6)
       
By using (x,z,u)=(4, 18, 7), we obtain parametric solution of (6).

x = 40k^2-546k+392, z = 180k^2+154k-294, u = -676k^2+938k-343.........(7)

Substitute (7) to (5), we obtain quartic elliptic curve
                 
(5y)^2 = 404400k^4+3071600k^3-5943700k^2+3704400k-960400.

Y^2 = X^3 - X^2 + 48152X, -22779024.

This rank is 1 and generator: P1=(9699595645/1397124, 955655258050783/1651400568).

Thus, we can obtain infinitely many integer solutions.

-P1:    (x,y,z,w)=(10617, 4728, 5306, 1210664898377)



Case 2: x^2+(u-z)x-zu-5z^2-6y^2=0..........................................(8)
       31x^2+(-35u+35z)x-6u^2-37zu+25z^2=0.................................(9)
       
x = 168k^2-949k+845, z = 246k^2-13k-338, u = -2731k^2+6201k-3549..........(10)

Substitute (10) to (8), we obtain quartic elliptic curve
                 
(6y)^2 = -615996k^4+12127752k^3-41438124k^2+52200720k-22620312.

Y^2 = X^3 - X^2 + 134232X + 35398076.

This rank is 2 and generators: P1=(-1082/9, 112976/27), P2=(1619054/7225, 5377751808/614125).

Thus, we can obtain infinitely many integer solutions.

P1:    (x,y,z,w)=(87036, 36221, 79758, 833297083257349)
P2:    (x,y,z,w)=(90, 85, 168, 4836493)
P1+P2: (x,y,z,w)=(29316, 13469, 5802, 25313949479269)


Case 3: 9x^2+(9u-9z)x-9zu-z^2-10y^2=0........................................(11)
       91x^2+(-19u+19z)x-90u^2-181zu+z^2=0...................................(12)
       
x = 9540k^2-3197k-1337, z = 5670k^2-11797k+3436, u = -18907k^2+22629k-6771...(13)

Substitute (13) to (11), we obtain quartic elliptic curve
                 
(10y)^2 = -3584015100k^4+855887400k^3+14458799300k^2-13812306400k+3364892600.

Y^2 = X^3 - X^2 - 973092648X + 10215887350716.

This rank is 2 and generators: P1=(197057918/5041, 2021818018000/357911), P2=(-161462099882/30569841, -659140288517995120/169020650889).

Thus, we can obtain infinitely many integer solutions.

 P1: (x,y,z,w)=(34396978040966046547551621140, 9839569329848106327963851187, 6004716786116987362108095390,
                40708580744275179990707676418405637465118464889690376838386300076404133626003366103797)
 P2: (x,y,z,w)=(2698323207050, 8596885845117, 8468726088480, 879189913172196898359105352457299716613)
-P1: (x,y,z,w)=(140, 213, 390, 60163597)





4.Reference


[1]. A. Bremner, M. Ulas, On x^a} y^b} z^c} w^d= 0, 1/a + 1/b + 1/c + 1/d = 1 , Int. J. NumberTheory, 7 (2011), no. 8



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