ILL 1983 Problem 18

Let $b \geq 2$ be a positive integer.

(a) Show that for an integer $N$ , written in base $b$ , to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits.

(b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$ , are equal to the sum of the squares of their digits.

(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.

(d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.