They key point is Faltings' proof (1983)

of the Mordell conjecture:

there are only finitely

many rational points on any

curve of genus greater than one.

So hunting for *integer* solutions to

(x^n-1)/(x-1)=y^2 for n>3

beyond the obvious

(x,n,y)=(7,4,20),(3,5,11)

is likely to be *very* unrewarding.

I do not recommend that folk spend

cycles on it. If you do, take

n>5, since Elkies must have

thrashed n=4,5 close to death.

David