46x^6 + 12y^6 - 58z^6 = w^2


We show diophantine equation 46x^6 + 12y^6 - 58z^6 = w^2 has infinitely many integer solutions.

46x^6 + 12y^6 - 58z^6 = w^2...........................................................(1)

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

Let (a,b,c)=(46, 12, -58),(p,q)=(2, -17/6) the we obtain

u^2 = 27155202816t^4-13923118080t^3+65005286400t^2+2212945920t+9958256640..............(2)

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

Y^2 = X^3-29682199953135X -30402432883650709350.

Rank is 2 and generator is (X,Y)=[-642574770/361 , -28106704645740/6859 ],
                                 [-3132817163641413504483946133639/836518915250101904642116 ,
                                  -4065287974064040900391114544515009468911899989/765091978035590053318991966979742664].

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

Numerical example:

[X,Y,Z,W] = [1, 28, 5, 76038].




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