39x^6 + 36y^6 - 75z^6 = w^2


We show diophantine equation 39x^6 + 36y^6 - 75z^6 = w^2 has infinitely many integer solutions.

39x^6 + 36y^6 - 75z^6 = w^2..........................................................................(1)

For details,ax^6 + by^6 + cz^6 = w^2 Ⅴ

Let (a,b,c)=(39, 36, -75),(p,q)=(2, -1/12) the we obtain

u^2 = 5269990061700t^4+15320192191200t^3+17925859224000t^2+10317041280000t+2652953472000..............(2)

This quartic equation is birationally equivalent to an elliptic curve below.

Y^2 = X^3-557601135X -2934790146730.

Rank is 3 and generator is (X,Y)=[53539668641/1993744 , 3397548506907121/2815166528],
                                 [-145461601/22801 , -2073912470098/3442951 ],
                                 [85258913393674599136957499/16178437439797808569 , -787236688439233365516317232536342924642/65073604106079354244957470253].

Hence we can obtain infinitely many integer solutions for equation (1).

Numerical example:

[X,Y,Z,W] = [279176, 136999, 79412, 136687662759694410].