An elliptic curve of the form for an integer. This equation has a finite number of solutions in integers for all nonzero . If is a solution, it therefore follows that is as well.
Uspensky and Heaslet (1939) give elementary solutions for , , and 2, and then give , , , and 1 as exercises. Euler found that the only integer solutions to the particular case (a special case of Catalan's conjecture) are , , and . This can be proved using Skolem's method, using the Thue equation , using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).
Solutions of the Mordell curve with are summarized in the table below for small .
solutions
1
(, 0), (0, 1), (2, 3)
2
(, 1)
3
(1, 2)
4
(0, 2)
5
(, 2)
6
none
7
none
8
(, 0), (1, 3), (2, 4), (46, 312)
9
(, 1), (0, 3), (3, 6), (6, 15), (40, 253)
10
(, 3)
Values of such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... (Sloane's A054504; Apostol 1976, p. 192).
for 
an integer. This equation has a finite number of solutions in integers for all nonzero 
. If 
is a solution, it therefore follows that 
is as well.
, 
, and 2, and then give 
, 
, 
, and 1 as exercises. Euler found that the only integer solutions to the particular case 
(a special case of Catalan's conjecture) are 
, 
, and 
. This can be proved using Skolem's method, using the Thue equation 
, using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).
are summarized in the table below for small 
.
, 0), (0, 1), (2, 3)
, 1)
, 2)
, 0), (1, 3), (2, 4), (46, 312)
, 1), (0, 3), (3, 6), (6, 15), (40, 253)
, 3)
such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... (Sloane's A054504; Apostol 1976, p. 192).