Primitive solutions of (5.1.4):
   order   x      y                    z
    1.    (144)  (133, 110, 84, 27)   61917364224   (Lander et al, 1967)
   No other solutions for 765 < x < 2834.
   Thomas Womack found no solutions for x < 32779.
 

Primitive solutions of (5.1.5):
   12 solutions for x.lt.600; see (Lander et al, 1967).
   A parametric solution is given by (Sastry, 1934).

Primitive solutions of (5.1.6):
   order   x      y
    1.    (12)  (11, 9, 7, 6, 5, 4)      (Lander et al, 1967)
    2.    (30)  (29, 19, 16, 11, 10, 5) 
    3.    (32)  (28, 24, 22, 17, 16, 15) 
    4.    (67)  (66, 36, 31, 23, 18, 13) 
    5.    (67)  (66, 34, 31, 29, 20, 7) 
    6.    (78)  (64, 61, 58, 48, 35, 22) 
    7.    (99)  (96, 67, 20, 19, 13, 4) 
    8.    (99)  (89, 73, 64, 60, 17, 6) 

Primitive solutions of (5.2.3):
   No solution for z<=8x10^12 (Lander et al, 1967).
   order   x         y
    ?.    (14132,220) (14068,6237,5027)   [found by Bob Scher and Michael Colvin - check this]

Primitive solutions of (5.2.4):
   order   x         y
    1.    (29, 3)   (28, 20, 10, 4)      (Lander et al, 1967)
          (38, 12)  (37, 25, 13, 5) 
          (52, 28)  (50, 35, 29, 26) 
          (64, 61)  (63, 62, 25, 5) 
          (85, 16)  (82, 53, 50, 6) 
          (96, 31)  (86, 72, 63, 56) 
          (97, 63)  (99, 37, 13, 11) 
          (99, 14)  (94, 67, 58, 44)
          (119, 41) (114, 76, 72, 48) 
          (127, 42) (112, 97, 93, 17)
          (127, 109) (137, 40, 35, 24)
          (134, 40)  (129, 92, 63, 10) 
          (144, 45)  (130, 113, 87, 69) 
          (156, 84)  (150, 105, 87, 78)
          (159, 84)  (135, 134, 109, 75) 
          (161, 127) (156, 134, 86, 62) 
          (166, 154) (174, 136, 83, 77) 
          (192, 3)   (172, 159, 93, 71) 
          (195, 60)  (191, 123, 52, 9) 
          (197, 96)  (196, 101, 84, 62)