42x^6 + 28y^6 - 70z^6 = w^2


We show diophantine equation 42x^6 + 28y^6 - 70z^6 = w^2 has infinitely many integer solutions.

42x^6 + 28y^6 - 70z^6 = w^2.......................................................................(1)

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

Let (a,b,c)=(42, 28, -70),(p,q)=(2, -1/2) the we obtain

u^2 = 1261799642880t^4+3681039559680t^3+4364119065600t^2+2529924096000t+758977228800..............(2)

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

Y^2 = X^3-59700375X -100238748750.

Rank is 3 and generator is (X,Y)=[9814 , -509012], [4884649/400 , -7966993643/8000], [-2061022675274/372837481 , 1775979476280548776/7199118920629].

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

Numerical example:

[X,Y,Z,W] = [367, 488, 25, 693386190].