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Subtract 1, get a square (Posted on 2009-06-24) Difficulty: 2 of 5
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    
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re: general solution (no proof yet) Comment 4 of 4 |
(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=n-2 y=2 and z=0 then
xyxyxyzy-1=
t^8-2t^7+3t^6-2t^5+3t^4-2t^3+2t^2+2=
(t^4-t^3+t^2+1)^2

thus x=n-2 y=2 and z=0 always gives a square plus 1

  Posted by Daniel on 2009-06-24 20:15:49

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