I searched the smallest solution of x3+by3+cz3=0.

Search condition
    b>=c,b>0,c>0
    b<50
    |y|<100000,|z|<100000
    xyz<>0
    smallest: min(max(|x|,|y|,|z|))

I made some filters to reduce the number of the search data.


    1. Consideration of x3+by3+cz3=0 mod 9

       In the case of (b,c) mod 9 =(4,2),(5,2),(7,4),(7,5),above equation have no solution.


    2. Consideration of x3+by3+cz3=0 mod 27

       In the case of (b,c) mod 27 =( 6, 3),( 9, 2),( 9, 3),( 9, 4),( 9, 5),( 9, 6),( 9, 7),
       (11, 4),(11, 5),(11, 9),(12, 3),(12, 6),(12, 9),(13, 7),(13, 9),(14, 7),(14, 9),(15, 3),
       (15, 6),(15, 9),(16, 9),(18, 2),(18, 3),(18, 4),(18, 5),(18, 6),(18, 7),(18,11),(18,12),
       (18,13),(18,14),(18,15),(18,16),(20, 4),(20, 5),(20, 9),(20,13),(20,14),(20,18),(21, 3),
       (21, 9),(21,12),(21,15),(21,18),(22, 7),(22, 9),(22,16),(22,18),(23, 7),(23, 9),
       (23,16),(23,18),(24, 6),(24, 9),(24,12),(24,15),(24,18),(24,21),(25, 9),(25,18),
       above equation have no solution.

    3. Consideration of x3+by3+cz3=0 mod 49
       In the case of (b,c) mod 49 =( 7, 2),( 7, 3),( 7, 4),( 7, 5),( 9, 7),
       (10, 7),(11, 7),(12, 7),(14, 2),(14, 3),(14, 4),(14, 5),(14, 7),(14, 9),(14,10),(14,11),
       (14,12),(16, 7),(16,14),(17, 7),(17,14),(18, 7),(18,14),(19, 7),(19,14),
       (21, 2),(21, 3),(21, 4),(21, 5),(21, 7),(21, 9),(21,10),(21,11),(21,12),(21,14),(21,16),
       (21,17),(21,18),(21,19),(23, 7),(23,14),(23,21),(24, 7),(24,14),(24,21),(25, 7),(25,14),
       (25,21),(26, 7),(26,14),(26,21),(28, 2),(28, 3),(28, 4),(28, 5),(28, 7),(28, 9),(28,10),
       (28,11),(28,12),(28,14),(28,16),(28,17),(28,18),(28,19),(28,23),(28,24),(28,25),(28,26),
       (30, 7),(30,14),(30,21),(30,28),(31, 7),(31,14),(31,21),(31,28),(32, 7),(32,14),(32,21),
       (32,28),(33, 7),(33,14),(33,21),(33,28),(35, 2),(35, 3),(35, 4),(35, 5),(35, 7),(35, 9),
       (35,10),(35,11),(35,12),(35,16),(35,17),(35,18),(35,19),(35,21),(35,23),(35,24),(35,25),
       (35,26),(35,28),(35,30),(35,31),(35,32),(35,33),(37, 7),(37,14),(37,21),(37,28),(37,35),
       (38, 7),(38,14),(38,21),(38,28),(38,35),(39, 7),(39,14),(39,21),(39,28),(39,35),
       (40, 7),(40,14),(40,21),(40,28),(40,35),(42, 2),(42, 3),(42, 4),(42, 5),(42, 9),(42,10),
       (42,11),(42,12),(42,14),(42,16),(42,17),(42,18),(42,19),(42,21),(42,23),(42,24),(42,25),
       (42,26),(42,28),(42,30),(42,31),(42,32),(42,33),(42,35),(42,37),(42,38),(42,39),(42,40),
       (44, 7),(44,14),(44,21),(44,28),(44,35),(44,42),(45, 7),(45,14),(45,21),(45,28),(45,35),
       (45,42),(46, 7),(46,14),(46,21),(46,28),(46,35),(46,42),(47, 7),(47,14),(47,21),(47,28),
       (47,35),(47,42),above equation have no solution.

   4. Consideration of x3+by3+cz3=0 mod 64
      In the case of (b,c) mod 64 =( 4, 2),( 6, 4),(10, 4),(12, 2),(12, 6),(12,10),(14, 4),
      (14,12),(18, 4),(18,12),(20, 2),(20, 6),(20,10),(20,14),(20,18),(22, 4),(22,12),(22,20),
      (26, 4),(26,12),(26,20),(28, 2),(28, 6),(28,10),(28,14),(28,18),(28,22),(28,26),
      (30, 4),(30,12),(30,20),(30,28),(34, 4),(34,12),(34,20),(34,28),(36, 2),(36, 6),(36,10),
      (36,14),(36,18),(36,22),(36,26),(36,30),(36,34),(38, 4),(38,12),(38,20),(38,28),(38,36),
      (42, 4),(42,12),(42,20),(42,28),(42,36),(44, 2),(44, 6),(44,10),(44,14),(44,18),(44,22),
      (44,26),(44,30),(44,34),(44,38),(44,42),(46, 4),(46,12),(46,20),(46,28),(46,36),(46,44)


   5.  Search for the rank of Y2=X3+(4bc)2
       By the Cohen's book[1.Cohen],there is a following relation between x3+by3+cz3=0
       and Y2=X3+(4bc)2.

       If the elliptic curve Y2=X3+(4bc)2 has zero rank,then the equation x3+by3+cz3=0
       has no nontrivial rational solutions.

       So,I computed the rank of Y2=X3+(4bc)2.
       As for the details,I computed the analytic rank as an approximate value of the algebraic rank.

       analytic rank = 0 ====>  algebraic rank = 0 ===> x3+by3+cz3=0 has only trivial solutions
       
   6.  Special case : x3+by3+bz3=0

       Trivial solution is (x,y,z)=(0,1,-1).
       Generally u3+v3=n  <====> E:Y2=X3-432n2
       Above case,(bu)3+(bv)3=-b2  <===> E:Y2=X3-432b^4

       We can find the points of elliptic curve E by Cremona's mwrank.
       Case b=30:

       One point is (X,Y)=( 373396/441,-148490056/9261).
       So,(u,v)=(-56068307/176429610,-18945793/176429610).
       Then, (176429610)3+30*(-56068307)3+30*(-18945793)3=0.
       
       In the same way
       Case b=22:
       (555170)3+22*(176021)3+22*(-236521)3=0

       Case b=34:
       (337755790019244956599422)3+34*(-103132220429179427450221)3+34*(-33116586760787233755671)3=0
       Too big!?




                       
                                       (x,y,z)=(-,-,-): locally not solvable  
                                       (x,y,z)=(?,?,?): solution not found 
                                       (x,y,z)=(0,0,0): only trivial solution 

                                       x3+by3+cz3=0