Diophantine equation $a^3+a=b^2+1$

Question

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?

Answer 1

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) $