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?

I used the following code

DATA "0","1","2","3","4","5","6","7","8","9"
DATA "A","B","C","D","E","F"
DIM d$(0 TO 15)
FOR i = 0 TO 15
 READ d$(i)
NEXT
FOR n = 3 TO 16
 FOR x = 1 TO n - 1
  FOR y = 0 TO n - 1
   IF y <> x THEN
    FOR z = 0 TO n - 1
     IF z <> x AND z <> y THEN
      num# = x * (n ^ 3 + n ^ 5 + n ^ 7) + y * (1 + n ^ 2 + n ^ 4 + n ^ 6) + z * n
      v# = num# - 1
      IF INT(SQR(v#)) = SQR(v#) THEN
       disp$ = "Base " + STR$(n) + ": "
       disp$ = disp$ + d$(x) + d$(y) + d$(x) + d$(y) + d$(x) + d$(y) + d$(z) + d$(y)
       PRINT disp$
      END IF
     END IF
    NEXT z
   END IF
  NEXT y
 NEXT x
NEXT n

 

and got the following answers

Base  3: 12121202
Base  5: 32323202
Base  6: 42424202
Base  7: 52525202
Base  8: 62626202
Base  9: 72727202
Base  10: 45454565
Base  10: 82828202
Base  11: 92929202
Base  12: A2A2A202
Base  13: B2B2B202
Base  14: C2C2C202
Base  15: D2D2D202
Base  16: E2E2E202

Posted by Daniel on 2009-06-24 12:52:23