1.Introduction

A. Bremner and M. Ulas([1]) gave 2 families of solutions of w^2 = x^6 + y^6 + z^6.
We show new 4 families of solutions of w^2 = x^6 + y^6 + z^6 by Bremner and Ulas's method.


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

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

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

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

(b*c+b^2+c^2)*x^2+((2*b*c+b^2)*u+(-2*b*c-b^2)*z)*x+(-b*c-c^2)*u^2+(-b^2-2*b*c-2*c^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: 10u^2+9ux+9uz+x^2-9xz-10y^2+z^2=0.................................(5)
       -90u^2+19ux+91x^2-181uz-19xz+z^2=0.................................(6)
       
By using (x,z,u)=(14,19,-39), we obtain parametric solution of (6).

x = 1260k^2+ 6115k+7255, z = 1710k^2+ 6925k+6890, u = 337k^2+1959k+2847...(7)

Substitute (7) to (5), we obtain quartic elliptic curve
                 
(10y)^2 = -47360000k^4-284280000k^3-372332500k^2+478560000k+937160000.

Y^2 = X^3 - X^2 - 755190920X + 7296213724400.

This rank is 2 and generators: P1=(195220 , 85439040) , P2=(2811180025396/20385225, 4625519785775320256/92039290875).

Thus, we can obtain infinitely many integer solutions.

P1:    (x,y,z,w)=(3878967192378, 2387315975324, 3144122041299, 67509800768409853596615289151070881941)
P2:    (x,y,z,w)=(4482, 3404, 3087, 102604114673 )
P2-P1: (x,y,z,w)=(462741, 29408, 502026, 160707356499029581 )



Case 2: 17u^2+3ux+3zu+14x^2-3xz-17y^2+14z^2=0...............................(8)
        247x^2+(280u-280z)x-51u^2-298zu+196z^2=0............................(9)

x = 714k^2+6122k+ 5502, z = 1377k^2+6356k+ 3383, u =872k^2+ 6020k+15974 ....(10)

Substitute (10) to (8), we obtain quartic elliptic curve

(17y)^2 = 835210000k^4+9816399998k^3+44582378654k^2+91904454766k+89961422222.

Y^2 = X^3 - X^2 -25532077917X -450714035035867.

This rank is 1 and generator: P1=(-179769559/10000, 1570048006989/1000000).

P1:(x,y,z,w)=(2111412001939505236688611146, 4657607627134818933308206100, 3600027763016113827334864353,
                  111688617976512089928288988687238176177504035790063668454499469660199535095816179905)
                  


Case 3: 19x^2+(15u-15z)x+34u^2+15zu+19z^2-34y^2=0.........................................(11)
        871x^2+(931u-931z)x-510u^2-1381zu+361z^2=0........................................(12)

x = 91800k^2-75274k + 10564, z = 78540k^2 -86974k+18390, u = -100684k^2 +113794k-32809....(13)

Substitute (13) to (11), we obtain quartic elliptic curve

(34y)^2 = 8723702969536k^4-18938620137152k^3+15896432677040k^2-6155914812544k+951363992192

Y^2 = X^3 - X^2 -1384251895768X -435570842878035984.

This rank is 2 and generators: P1=(-437670 , 294001026) , P2=(-883199259630/1585081, 805044959957239914/1995616979).

P1:    (x,y,z,w)=(16010211756, 18121673282, 20918940507, 11664288725635670476866500965373)
P2:    (x,y,z,w)=(2184, 2518, 2043, 20883327517)
P1+P2: (x,y,z,w)=(1060578960, 1717246918, 1256489295, 5568022893866546525064082993 )


Case 4: 24x^2+(u-z)x+25u^2+zu+24z^2-25y^2=0........................................(14)
        601x^2+(624u-624z)x-25u^2-626zu+576z^2=0...................................(15)


x = 900k^2-22454k+35048, z = 1625k^2-22396k+19011, u = -20376k^2+55128k-47558......(16)

Substitute (16) to (14), we obtain quartic elliptic curve

(25y)^2 = 260235937500k^4-1444290031250k^3+3682342958750k^2-4574383836250k+2286539471250

Y^2 = X^3 - X^2 -1061616758925X -266880548537456875.

This rank is unknown, one solution is P1=(-7099797566707981919843/23609015159280516, 575597937755251598521628384145847/3627578393943620301683064).

By the Nagell-Lutz theorem, this is not a point of finite order.

P1:    (x,y,z,w)=(460796649, 768261270, 236413180, 464071352006298597642683801)



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