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?
(this is my first posting, hope it goes well)
There can be no such (integer) base.
Consider using base b. 2005 base b = 2*b^3 + 5.
If this would be a square, we would have 2*b^3+5=a^2 for some integer
a. The left-hand side is odd, so the integer a must also be odd: a =
2*a' +1.
this gives
2*b^3+5 = 4*a'^2 +4*a'+1
2*b^3+4 = 4*a'^2 + 4*a'
b^3 = 2*a'^2+2*a'-2
This means b must be even : b = 2*b'
8*b' = 2*a'^2 + 2*a' -2
4*b' = a'^2 + a' -1
The left hand side will always be even, the right hand side will always be odd, therefore there can be no solution.
Solution
