MathDB

Given a positive integer

n

, it can be shown that every complex number of the form

r+si

, where

r

and

s

are integers, can be uniquely expressed in the base

-n+i

using the integers

1,2,\ldots,n^2

as digits. That is, the equation

r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0

is true for a unique choice of non-negative integer

m

and digits

a_0,a_1,\ldots,a_m

chosen from the set

\{0,1,2,\ldots,n^2\}

, with

a_m\ne 0

. We write

r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}

to denote the base

-n+i

expansion of

r+si

. There are only finitely many integers

k+0i

that have four-digit expansions

k=(a_3a_2a_1a_0)_{-3+i}~~~~a_3\ne 0.

Find the sum of all such

k

Problem Statement