1.Introduction

I searched the solutions of ax^6 + by^3 = cz^2. 

First,we transform the equation (1) to the elliptic curve (2).

Divide both sides of (1) by x^6 and set X=(bcy)/x^2, Y=(bc^2z)/x^3, then we obtain (2).

ax^6 + by^3 = cz^2.....................................................(1).

Y^2 = X^3 + ab^2c^3....................................................(2).



If (2) has rank zero, solutions are given below.

(a). ab^2c^3= D^2
     Solution is (X,Y)=(0, +-D).
     (x,y,z)=(+-bc^2, 0, +-Db^2c^4).

(b). ab^2c^3= D^3
     Solution is (X,Y)=(-D, 0).
     (x,y,z)=(+-bc, -Dbc,0).

(c). ab^2c^3= D^6
     Solution are (X,Y)=(0, +-D^3),(-D^2, 0),(2D^2, +-3D^3).
     (x,y,z)=(bc^2, 0, D^3b^2c^4),(+-bc, -D^2bc, 0),(+-bc^2, 2D^2bc^3, +-3D^3b^2c^4).

(d). ab^2c^3= -432*D^6
     Solution are (X,Y)=(12D^2, +-36D^3),(+-bc^2, 12D^2bc^3, +-36D^3b^2c^4).

(e). Otherwise
     No solution( [1].Cohen. P.397).

If (2) has rank nonzero, (2) has infinitely many solutions( [1].Cohen. P.397).




2. Search results

I searched the solutions of (1) under the condition  [a, b, c] < 5.
Search results.

       RK       : Rank of (2).
       
       Case: [1]: Case (a).
             [2]: Case (b).
             [3]: Case (c).

       Solutions: Smallest solution(If (2) has nonzero rank).
                  NONE:No solution.

[a, b, c]  RK  Case          Solutions:[x, y, z]

[1, 1, 1] [0] [3] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[1, 1, 1] [0] [3] [[1, -1, 0],[-1, -1, 0]]
[1, 1, 2] [1] [2] [1, -1, 0]
[1, 1, 3] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[1, 1, 4] [0] [3] [[1, -1, 0],[-1, -1, 0]]
[1, 1, 4] [0] [3] [[2, 0, 4], [2, 0, -4], [-2, 0, 4], [-2, 0, -4]]
[1, 1, 5] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[1, 2, 1] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[1, 2, 2] [0] [ ] [NONE]
[1, 2, 3] [1] [ ] [1, 1, 1]
[1, 2, 4] [0] [1] [[2, 0, 4], [2, 0, -4], [-2, 0, 4], [-2, 0, -4]]
[1, 2, 5] [1] [ ] [2, 2, 4]
[1, 3, 1] [1] [1] [1, 0, 1]
[1, 3, 2] [1] [ ] [3, -3, 18]
[1, 3, 3] [0] [ ] [NONE]
[1, 3, 4] [1] [1] [1, 1, 1]
[1, 3, 5] [0] [ ] [NONE]
[1, 4, 1] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[1, 4, 2] [1] [ ] [2, -2, 4]
[1, 4, 3] [0] [ ] [NONE]
[1, 4, 4] [0] [1] [[2, 0, 4], [2, 0, -4], [-2, 0, 4], [-2, 0, -4]]
[1, 4, 5] [1] [ ] [1, 1, 1]
[1, 5, 1] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[1, 5, 2] [0] [ ] [NONE]
[1, 5, 3] [1] [ ] [15, 15, 1950]
[1, 5, 4] [0] [1] [[2, 0, 4], [2, 0, -4], [-2, 0, 4], [-2, 0, -4]]
[1, 5, 5] [0] [ ] [NONE]
[2, 1, 1] [1] [ ] [1, -1, 1]
[2, 1, 2] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[2, 1, 3] [1] [ ] [1, 1, 1]
[2, 1, 4] [1] [ ] [2, -4, 4]
[2, 1, 5] [0] [ ] [NONE]
[2, 2, 1] [1] [2] [1, -1, 0]
[2, 2, 3] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[2, 2, 5] [1] [2] [1, -1, 0]
[2, 3, 1] [1] [ ] [3, 21, 171]
[2, 3, 2] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[2, 3, 3] [1] [ ] [3, -5, 19]
[2, 3, 4] [1] [ ] [6, 84, 684]
[2, 3, 5] [2] [ ] [1, 1, 1]
[2, 4, 1] [0] [ ] [NONE]
[2, 4, 3] [0] [ ] [NONE]
[2, 4, 5] [1] [ ] [2, -3, 2]
[2, 5, 1] [1] [ ] [5, -5, 175]
[2, 5, 2] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[2, 5, 3] [1] [ ] [3, -3, 21]
[2, 5, 4] [1] [ ] [10, -20, 700]
[2, 5, 5] [1] [ ] [125, 42975, 8994125]
[3, 1, 1] [1] [ ] [1, 1, 2]
[3, 1, 2] [2] [ ] [1, -1, 1]
[3, 1, 3] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[3, 1, 4] [1] [ ] [1, 1, 1]
[3, 1, 5] [0] [ ] [NONE]
[3, 2, 1] [1] [ ] [1, -1, 1]
[3, 2, 2] [0] [ ] [NONE]
[3, 2, 3] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[3, 2, 4] [1] [ ] [2, -4, 4]
[3, 2, 5] [2] [ ] [1, 1, 1]
[3, 3, 1] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[3, 3, 2] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[3, 3, 4] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[3, 3, 5] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[3, 4, 1] [1] [ ] [2, 1, 14]
[3, 4, 2] [0] [ ] [NONE]
[3, 4, 3] [1] [1] [1, 0, 1]
[3, 4, 4] [1] [ ] [2, 1, 7]
[3, 4, 5] [0] [ ] [NONE]
[3, 5, 1] [0] [ ] [NONE]
[3, 5, 2] [1] [ ] [1, 1, 2]
[3, 5, 3] [1] [1] [1, 0, 1]
[3, 5, 4] [0] [ ] [NONE]
[3, 5, 5] [0] [ ] [NONE]
[4, 1, 1] [0] [1] [[1, 0, 2], [1, 0, -2], [-1, 0, 2], [-1, 0, -2]]
[4, 1, 2] [0] [ ] [NONE]
[4, 1, 3] [1] [ ] [1, -1, 1]
[4, 1, 4] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[4, 1, 5] [1] [ ] [1, 1, 1]
[4, 2, 1] [0] [1] [[1, 0, 2], [1, 0, -2], [-1, 0, 2], [-1, 0, -2]]
[4, 2, 3] [0] [ ] [NONE]
[4, 2, 5] [1] [ ] [1, 2, 2]
[4, 3, 1] [1] [1] [1, -1, 1]
[4, 3, 2] [1] [ ] [6, 6, 306]
[4, 3, 3] [0] [ ] [NONE]
[4, 3, 4] [1] [1] [1, 0, 1]
[4, 3, 5] [0] [ ] [NONE]
[4, 4, 1] [0] [3] [[1, 0, 2], [1, 0, -2], [-1, 0, 2], [-1, 0, -2]]
[4, 4, 1] [0] [3] [[1, -1, 0],[-1, -1, 0]]
[4, 4, 3] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[4, 4, 5] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[4, 5, 1] [1] [1] [1, 0, 2]
[4, 5, 2] [0] [ ] [NONE]
[4, 5, 3] [1] [ ] [5, 10, 150]
[4, 5, 4] [1] [1] [1, 0, 1]
[4, 5, 5] [0] [ ] [NONE]
[5, 1, 1] [1] [ ] [1, -1, 2]
[5, 1, 2] [1] [ ] [1, 3, 4]
[5, 1, 3] [0] [ ] [NONE]
[5, 1, 4] [1] [ ] [1, -1, 1]
[5, 1, 5] [1] [1] [1, 0, 1]
[5, 2, 1] [0] [ ] [NONE]
[5, 2, 2] [0] [ ] [NONE]
[5, 2, 3] [1] [ ] [1, -1, 1]
[5, 2, 4] [0] [ ] [NONE]
[5, 2, 5] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[5, 3, 1] [0] [ ] [NONE]
[5, 3, 2] [2] [ ] [1, -1, 1]
[5, 3, 3] [1] [ ] [21, 1009, 34208]
[5, 3, 4] [0] [ ] [NONE]
[5, 3, 5] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[5, 4, 1] [1] [ ] [1, -1, 1]
[5, 4, 2] [1] [ ] [2, -2, 12]
[5, 4, 3] [0] [ ] [NONE]
[5, 4, 4] [1] [ ] [2, -4, 4]
[5, 4, 5] [0] [1] [[1, 0, 1], [1, 0, -1], [-1, 0, 1], [-1, 0, -1]]
[5, 5, 1] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[5, 5, 2] [1] [2] [1, -1, 0]
[5, 5, 3] [0] [2] [[1, -1, 0], [-1, -1, 0]]
[5, 5, 4] [0] [2] [[1, -1, 0], [-1, -1, 0]]

3. References

[1].Henri Cohen:Number Theory Volume 1:Tools and Diophantine Equations.



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