MathDB

Unique Rational Number Representation

Source: ISI 2021 P3

July 18, 2021

number theoryalgebrapower seriesisi

Problem Statement

Prove that every positive rational number can be expressed uniquely as a finite sum of the form

a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},

where

a_n

are integers such that

0 \leq a_n \leq n-1

for all

n > 1