ax^6 + by^6 + cz^6 = w^2


We show diophantine equation ax^6 + by^6 + cz^6 = w^2 has infinitely many integer solutions.

ax^6 + by^6 + cz^6 = w^2........................................................(1)

Let x=pt+1, y=qt+1, z=t+1, c = -a-b, q = (a+b-ap)/b then we obtain

v^2 = a(p^6b^5+5b^5+a^5-30ba^4p-60b^2a^3p+60ba^4p^2-60b^3a^2p+90b^2a^3p^2-60ba^4p^3-30b^4ap+60b^3a^2p^2-60b^2a^3p^3+30ba^4p^4+15ab^4+6a^4b-6a^5p+15a^3b^2+15a^5p^2+20a^2b^3-20a^5p^3+15a^5p^4-6a^5p^5+a^5p^6-6b^5p+15b^4ap^2-20b^3a^2p^3+15b^2a^3p^4-6ba^4p^5)bt^4
    + a(-30ba^4p-120b^2a^3p-180b^3a^2p-60b^3a^2p^3+30b^2a^3p^4+60a^2b^3+60ab^4+60b^4ap^2+30a^3b^2+6a^4b+30ba^4p^4-6ba^4p^5+180b^2a^3p^2-120b^4ap+180b^3a^2p^2-120b^2a^3p^3-30b^5p+60ba^4p^2-60ba^4p^3+24b^5+6b^5p^5)bt^3
    + a(15b^2a^3p^4-60b^2a^3p+90b^2a^3p^2+60a^2b^3-180b^3a^2p-60b^3a^2p^3-60b^2a^3p^3-60b^5p+180b^3a^2p^2+15b^5p^4+90ab^4-180b^4ap+90b^4ap^2+45b^5+15a^3b^2)bt^2
    + a(60b^4ap^2-20b^3a^2p^3-60b^3a^2p+60b^3a^2p^2-120b^4ap-60b^5p+20b^5p^3+20a^2b^3+60ab^4+40b^5)bt
    + a(15b^5p^2+15ab^4-30b^5p+15b^5-30b^4ap+15b^4ap^2)b........................(2)

a,b,p are arbitrary.

This quartic equation has infinitely many rational solutions for [a,b,c] = [6, 4, -10],[9, 3, -12],[10, 2, -12],[10, 6, -16],
[12, 3, -15],[12, 6, -18],[15, 5, -20][16, 2, -18],[16, 9, -25],[20, 5, -25],[20, 10, -30],[21, 7, -28],[24, 16, -40],[25, 20, -45],
[27, 12, -39],[28, 14, -42],[30, 6, -36],[30, 20, -50],[36, 12, -48],[38, 10, -48],[38, 20, -58],[39, 13, -52],[39, 36, -75],
[40, 8, -48],[40, 24, -64],[42, 28, -70],[46, 12, -58],[48, 24, -72] with (a,b,c)<50.

For details, please refer to each (a,b,c) below.

Hence we can obtain infinitely many integer solutions for equation (1) for above (a,b,c).

Example:

[a, b, c]    [p, q]

[6, 4, -10]   [2, -1/2]  [6x^6 + 4y^6 - 10z^6   = w^2]
[9, 3, -12]   [2, -2]    [9x^6 + 3y^6 - 12z^6   = w^2]
[10, 2, -12]  [2, -4]    [10x^6 + 2y^6 - 12z^6  = w^2]
[10, 6, -16]  [2, -2/3]  [10x^6 + 6y^6 - 16z^6  = w^2]
[12, 3, -15]  [2, -3]    [12x^6 + 3y^6 - 15z^6  = w^2]
[12, 6, -18]  [2, -1]    [12x^6 + 6y^6 - 18z^6  = w^2]
[15, 5, -20]  [2, -2]    [15x^6 + 5y^6 - 20z^6  = w^2]
[16, 2, -18]  [2, -7]    [16x^6 + 2y^6 - 18z^6  = w^2]
[16, 9, -25]  [2, -7/9]  [16x^6 + 9y^6 - 25z^6  = w^2]
[20, 5, -25]  [3, -7]    [20x^6 + 5y^6 - 25z^6  = w^2]
[20, 10, -30] [2, -1]    [20x^6 + 10y^6 - 30z^6 = w^2]
[21, 7, -28]  [2, -2]    [21x^6 + 7y^6 - 28z^6  = w^2]
[24, 16, -40] [2, -1/2]  [24x^6 + 16y^6 - 40z^6 = w^2]
[25, 20, -45] [2, -1/4]  [25x^6 + 20y^6 - 45z^6 = w^2]
[27, 12, -39] [2, -5/4]  [27x^6 + 12y^6 - 39z^6 = w^2]
[28, 14, -42] [2, -1]    [28x^6 + 14y^6 - 42z^6 = w^2]
[30, 6, -36]  [2, -4]    [30x^6 + 6y^6 - 36z^6  = w^2]
[30, 20, -50] [2, -1/2]  [30x^6 + 20y^6 - 50z^6 = w^2]
[33, 3, -36]  [2, -10]   [33x^6 + 3y^6 - 36z^6  = w^2]
[36, 12, -48] [2, -2]    [36x^6 + 12y^6 - 48z^6 = w^2]
[38, 10, -48] [2, -14/5] [38x^6 + 10y^6 - 48z^6 = w^2]
[38, 20, -58] [2, -9/10] [38x^6 + 20y^6 - 58z^6 = w^2]
[39, 13, -52] [2, -2]    [39x^6 + 13y^6 - 52z^6 = w^2]
[39, 36, -75] [2, -1/12] [39x^6 + 36y^6 - 75z^6 = w^2]
[40, 8, -48]  [2, -4]    [40x^6 + 8y^6 - 48z^6  = w^2]
[40, 24, -64] [2, -2/3]  [40x^6 + 24y^6 - 64z^6 = w^2]
[42, 28, -70] [2, -1/2]  [42x^6 + 28y^6 - 70z^6 = w^2]
[45, 20, -65] [2, -5/4]  [45x^6 + 20y^6 - 65z^6 = w^2]
[46, 12, -58] [2, -17/6] [46x^6 + 12y^6 - 58z^6 = w^2]
[48, 12, -60] [2, -3]    [48x^6 + 12y^6 - 60z^6 = w^2]
[48, 24, -72] [2, -1]    [48x^6 + 24y^6 - 72z^6 = w^2]