The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.


Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? 


Solution: As part of my regular involvement on this problem, I have found the following set of integer solutions by trail and error method or inspection method.


i.e., $(0, -1), (1, -1), (0, 0) =$ trivial solution, $(1, 0),$ $(0, 1)$ and $(1, 1)$.


If there exits still more solutions apart from above, HOW TO FIND SUCH SOLUTIONS? Importantly, is there any method to find solutions instead of trail or inspection method?


with regards,
mahoob



Special type Diophantine equations with integer solutions
If one could prove the explicit $abc$-conjecture, i.e., with an explicit constant $C$, then one would have explicit bounds on the heights of the rational points in Falting's theorem. If this is not available, one can try other methods. For example, consider the Fermat quartics $x^4+y^4=c$. For $c=17$ this is Serre's curve. Serre asked whether or not there are rational solutions other than the obvious ones $(\pm 1,\pm 2),(\pm 2,\pm 1)$. This could be proved by Flyn in $2001$ using Chabauty techniques. All other cases $c\le 81$ can be solved easier, i.e., by local methods or by a map onto
an elliptic curve of rank $0$.


All integer solutions of $y^2-y=x^5-x$ have beed determined by Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely in $2008$. 
They are
$$
(x,y)=(-1,0),(-1,1),(0,0),(0,1),(1,0),(1,1),(2,-5),(2,6),(3,-15),(3,16),(30,-4929), (30,4939).
$$


