MathDB

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.

Problem Statement