How to solve special type of Diophantine equation

Question

I am so excited to learn finding integer solutions of the equation $x^2 -y^5 = x-y$. I just found few solutions by plugging various integers in place of $x$ and $y$. But, I need a permanent method or approach to find all most all OR as much as we can "integer" solutions of the cited above equation. Kindly help. with regards Pokwishi

Answer 1

A routine in Pari/GP:

{gettime();print(" ");
 y_max = 10 000 000; 
 for(y = 0, y_max,
     w = y^5-y; 
             \\  x_lo=floor(sqrt(w +1/4)+1/2); x_hi=x_lo+1; \\ originally floor & ceil
         x_lo=sqrtint(w)+1;x_hi=x_lo+1;                     \\ this is more efficient
     w_lo=x_lo*(x_lo-1);  if(w_lo==w ,print("y=",y," x=",x_lo," (",w_lo," = ", w,")");next());
     w_hi=x_hi*(x_hi-1);  if(w_hi==w ,print("y=",y," x=",x_hi," (",w_hi," = ", w,")");next());
     );
   print(gettime()," msec") ;
   print("(no more solutions up to y=",y_max,")");  
 }   

\\ output: =================================================

y=0 x=1 (0 = 0)
y=1 x=1 (0 = 0)
y=2 x=6 (30 = 30)
y=3 x=16 (240 = 240)
y=30 x=4930 (24299970 = 24299970)
51995 msec
(no more solutions up to y=10 000 000)