1989 AIME Problems/Problem 14

Problem

Given a positive integer , it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as digits. That is, the equation

is true for a unique choice of non-negative integer and digits chosen from the set , with $a_m\ne 0^{}^{}$ (Error compiling LaTeX. Unknown error_msg). We write

to denote the base expansion of . There are only finitely many integers that have four-digit expansions

Find the sum of all such .

Solution

First, we find the first three powers of :

So we need to solve the diophantine equation .

The minimum the left hand side can go is -54, so , so we try cases:

• Case 1:

The only solution to that is .

• Case 2:

The only solution to that is .

• Case 3:

cannot be 0, or else we do not have a four digit number.

So we have the four digit integers and , and we need to find the sum of all integers that can be expressed by one of those.

:

We plug the first three digits into base 10 to get . The sum of the integers in that form is .

:

We plug the first three digits into base 10 to get . The sum of the integers in that form is . The answer is .