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 d2 < b, then the three digit number (d2)(2d2)(d2) has value d2(b+1)2. So take N to be any number divisible by 1, 2, 3, ... , 2002 and greater than 20022. Then if we write N2 in base N/d - 1 for d = 1, 2, 3, ... , 2002 we get the three-digit palindrome (d2)(2d2)(d2).
© John Scholes
jscholes@kalva.demon.co.uk
11 Dec 2002