MathDB

Numbers in base (1+i)

Source: Iran 3rd round 2009 - final exam problem 6

January 2, 2015

inductionalgorithmfunctiontrigonometrynumber theory unsolvednumber theory

Problem Statement

Let

z

be a complex non-zero number such that

Re(z),Im(z)\in \mathbb{Z}

. Prove that

z

is uniquely representable as

a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n

where

n\geq 0

and

a_j \in \{0,1\}

and

a_n=1

.
Time allowed for this problem was 1 hour.