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?
Considering base b, we seek 2b³ + 5 = a², for some positive integers a, b.
By inspection, modulo 8, 2b³ + 5 = 3, 5, or 7, while a² = 0, 1, or 4.
Hence there are no solutions; 2005 is not a square in any integer base.
(A motivation for working modulo 8 is that a must be odd, and therefore a² = 1 mod 8. We hope that 2b³ + 5 never equals 1 mod 8.)
Modulo solution
