Here are two points,P (-1,4), P (2,5)
Which can be easily verified by substituting them into E.
Found some more points, (-2,3), (4,9), (8,23), (43,282), (52,375) So we have a total of 7 points.
Maybe someone could lend a hand by implementing
for i in range(-3,n):
x = sqrt(i^3 + 17)
if x == int
To see if there are any more values, (I tried to put this code into SAGE but doesn't seem to work, could anyway tell me why not, and how I could get it working?) I will ask my computer science friend if not =)
Oooh I skimmed past the first 10000 points, must of missed that one, good job. Did you write a program for that? The gap seems to be getting very large between the points, will be interesting to how many more there are! Any idea how to write a program in SAGE that would find such points?
Thanks for taking an interest!
That is a neat technique for finding such points, but of course it is not going to help in proving that there are only finitely many of them.
Edit. I just came across this reference to a theorem of Siegel. It also states that the eight integer solutions to y^2=x^3+17 are the only ones with y>0.