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 N-1, and X is nonzero.

What are the integer value(s) of N, with 3 ≤ N ≤ 16 for which this is possible?

in exploring further bases up to n=36 I found a pattern among solutions, namely

x=n-2 y=2 z=0 always seems to give a perfect square plus 1 for all bases n.  Of course in this particular problem n=4 must be excluded because then x=y=2.  I am currently working on a proof of this, but the most obvious aproach of trying to factor the polynomial given by xyxyxyzy when expanded out in base n bore no fruit.

Edited on June 24, 2009, 2:45 pm

Posted by Daniel on 2009-06-24 14:43:38