A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2.

ant101
Sep 4, 2018

#1**+1 **

assuming you're not counting 0....

4 is the smallest 5 digit base 2 palindrome, 10 is the 2nd smallest.

in base 3 10 is the length 3 palindrome 101

Rom
Sep 5, 2018

#4**+1 **

There are only 5 5 digit palindromes in base 2. They are

10001 base 2 which equals 17 base 10

10101 base 2 which equals 21 base 10

11011 base 2 which equals 27 base 10

11111 base 2 which equals 31 base 10

Which of these can be expressed as a palindrome in any base? Let the base be x

which is of the form ax^2+bx+a (base 10)

where a b and x are all integers

and a and b are both smaller than x

x=2 is too small

x=3, a=2 9a+b+a = 10a+b = 20+1 works 212 (base3) = 10101 (base2) = 21 (base 10)

So the question becomes... can I make 17 with a 3 digit palindrome.

**101 base 4 That is obvious I should have seen it in the first place**

\(10001_2=17_{10}=101_4\)

So the answer is 10001

Melody
Sep 9, 2018