TITLE: Diophantine equation $a^3+a=b^2+1$
QUESTION [5 upvotes]: I have this Diophantine:
$$ a^3+a=b^2+1 $$
I found $a=2$, $b=3$ works.
Also $a=13$ , $b=47$ works.
How can I find all the integer solutions?

REPLY [4 votes]: Using MAGMA and the method described by Pete L. Clark here one finds the complete set of integer solutions is 
$(1 , -1), (2 , 3), (13, -47), (1 , 1), (2 , -3), (13 , 47)
$