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 Subtract 1, get a square (Posted on 2009-06-24)
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?

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution | Comment 1 of 4

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
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

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

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