2005 base 10 is not a square. Neither is 2005 base 7 a square (equal 2*7^3+5=691). Is there any base b such that 2005 base b is a square?

(In reply to Since it was not tacitily stated... by pcbouhid)

Right.  "Square" should have been "square of an integer".  However, the problem becomes trivial without the implicit restriction; e.g., 2005=(sqr(2005))^2.

Wierd number representations, like your negative base, b=-3, have some interesting applications.  For example, using base b=-2 and "digits" 0, 1, w, w^2, where w is a complex third root of unity, you get a proper subset of all complex numbers (not every complex number has such a representation).  Those with norm less than 1 turn out to form a very attractive fractal.

Posted by McWorter on 2005-06-04 23:53:17