By subtracting 1 from the positive base N integer having the form XYXYXYZY, we get a perfect square. It is known that each of X, Y and Z represents a different base N digit from 0 to N1, and X is nonzero.
What are the integer value(s) of N, with 3 ≤ N ≤ 16 for which this is possible?
(In reply to
general solution (no proof yet) by Daniel)
I made a slight mistake in my initial attempt at a proof, so here is the proof.
if x=n2 y=2 and z=0 then
xyxyxyzy1=
t^82t^7+3t^62t^5+3t^42t^3+2t^2+2=
(t^4t^3+t^2+1)^2
thus x=n2 y=2 and z=0 always gives a square plus 1

Posted by Daniel
on 20090624 20:15:49 