A base b palindrome is an integer which is the same when read backwards in base b. For example, 200 is not a palindrome in base 10, but it is a palindrome in base 9 (242) and base 7 (404). Show that there is an integer which for at least 2002 values of b has three digits and is a palindrome.

**Solution**

Note that 121 has value (b+1)^{2} in base b. So if d^{2} < b, then the three digit number (d^{2})(2d^{2})(d^{2}) has value d^{2}(b+1)^{2}. So take N to be any number divisible by 1, 2, 3, ... , 2002 and greater than 2002^{2}. Then if we write N^{2} in base N/d - 1 for d = 1, 2, 3, ... , 2002 we get the three-digit palindrome (d^{2})(2d^{2})(d^{2}).

© John Scholes

jscholes@kalva.demon.co.uk

11 Dec 2002