Show how to find a number that is a b-palindrome, of at least three digits, for at least 1000 different values of b.

For example, 200 is not a 10-palindrome, but it is a 9-palindrome (242) and a 7-palindrome (404).

Those formulae work, but the numbers are enormous .. my intended answer yields a somewhat large number as well, but on a substantially smaller order.

For example, say you were looking for a number that worked for just 7 values of b, instead of 1000. 2^(7!) is 1518 digits long, 2^(2^8)+2^(2^7+1) is 78 digits long, but mine is only three digits long in decimal.

So, while your answers are perfectly fine, there is still much room for optimization. Remember, if you are in, say, base 25, a value of 22 is still a single digit (even though we don't have a way to represent it as such with the current system). In other words, if you want an ideal solution, your palindromes will probably contain more than just zeroes and ones.