On Some Properties of Digital Roots

Digital roots of numbers have several interesting properties, most of which are well-known. In this paper, our goal is to prove some lesser known results concerning the digital roots of powers of numbers in an arithmetic progression. We will also state some theorems concerning the digital roots of Fermat numbers and star numbers. We will conclude our paper by an interesting application.


Introduction
The part of mathematics that deals with the properties of specific types of numbers and their uses in puzzles and recreational mathematics has always fascinated scientists and mathematicians (O'Beirne 1961 [1], Gardner 1987 [2]).
In this short paper, we will talk about digital roots-a well-established and useful part of recreational mathematics which materializes in as diverse applications as computer programming (Trott 2004) [3] and numerology (Ghannam 2012 [4]).As will see, digital roots are equivalent to modulo 9 arithmetic (Property 1.6) and hence can be thought of as a special case of modular arithmetic of Gauss (Dudley, 1978) [5].
Let us start out by the following existence theorem: Theorem 1.1.Let n be a natural number and let ( ) s n denote the sum of the digits of n .In a finite num- ber of steps, the sequence ( ) ( , , , s n s s n s s s n  becomes a constant.Proof.Let , where for any 0 j k ≤ ≤ , 0 9 j d ≤ ≤ .This implies that ( ) is the required constant.
Else, at least one of

( )
s n n < Thus, repeatedly applying the s operator, we will get a decreasing sequence of numbers.Once a term of this sequence becomes a single digit number, from then on the sequence will remain a constant.Here are some simple properties that follow immediately from this definition: Property 1.1.( ) , where x     stands for the geatest integer less than or equal to x .
Property 1.2.( ) Property 1.6.This 9 9 × symmetric matrix Table 1, which is formed by replacing the numbers in a regular 9 9 × multiplication table by their digital roots, is referred to as a Vedic square.Vedic squares have been used extensively to create geometric patterns and symmetries, and even musical compositions by highlighting specific numbers.For more information see Pritchard (2003) [6].
Closely related to the concept of digital roots is that of additive persistence, which is defined as the number of (additive) steps required to obtain its digital root.We will denote the additive persistence of a nonnegative integer n by ( ) AP n .Clearly, for any single digit number the additive persistence is 1.
( ) k 's.For more information on additive persistence see Hinden (1974) [7].Some Well-Known Results Proposition 1.1.Digital root of a square is 1, 4, 7, or 9.By Property 1.2, the digital root of 2 x is ( ) Proof is similar to the one given above.Proposition 1.

Digital roots of the powers of a natural number x form a cyclical sequence. This cycle is the same for all numbers 9 x k + , where k is any natural number:
This follows because for any x , 0 9 x ≤ ≤ and for any two natural numbers k and r ( ) We can use Here, the last equality follows from properties 1, 2, and 3 above.By proposition 1.2, ( ) 4, 7,1, 7,1, In the following sections, we will prove some results on digital roots of powers of numbers in an arithmetic progression as well as digital roots of Fermat numbers and star numbers.

Digital Roots of Powers of Numbers in an Arithmetic Progression
We start with the following Proposition 2.1.Let k, m and n be three consecutive terms in an arithmetic progression with common difference d .Let Using the fact that a sum is divisible by a positive integer if all terms are divisible by a positive integer we get Theorem 2.1.Let q be a multiple of three.Let 1 2 , , , q n n n  be any q consecutive terms of an arithmetic progression whose common difference d is not a multiple of three.Let x n n n = + + +  Then, ( )  ( ) ρ = .Corollary 2.1.Let q be a multiple of three.Putting 1 d = , we get that the sum of the cubes of q consecutive integers is divisible by 9. Putting 2 d = , we get that the sum of the cubes of q consecutive odd integers (even integers) is divisible by 9.
Although similar results do not necessarily hold for sixth powers, we show that they do for ninth powers.In fact, we find out that the restriction on d is not needed for ninth powers.

9
, , , n n n  be nine consecutive terms in an arithmetic progression with common difference d.Let x n n n = + + +  Then, ( ) 9 x ρ = .Proof.This follows by writing Using the fact that a sum is divisible by a positive integer if all terms are divisible by a positive integer we get Theorem 2.2.Let q be a multiple of nine.Let 1 2 , , , q n n n  be any q consecutive terms of an arithmetic progression whose common difference d.Let x n n n = + + +  Then, ( ) 9 x ρ = .Corollary 2.2.Let q be a multiple of nine.Putting 1 d = , we get that the sum of the ninth powers of q consecutive integers is divisible by 9. Putting 2 d = , we get that the sum of the ninth powers of q consecutive odd integers (even integers) is divisible by 9.

Digital Roots of Fermat Numbers
As is well-known, a Fermat number n F is defined as  ( )

Digital Roots of Star Numbers
The th j star number (so called because geometrically these numbers can be arranged to represent hexagrams) is denoted as j s and is of the form ( )

Definition 1 . 1 .
Let ( ) follows.To generalize the concept of digital roots to any other base b , one should simply change the 9 in the formulas to 1 b − .For more information on digital roots seeAverbach and Chein (2000)  [8].
If d is not a multiple of 3, then ( ) follows.Inspection of the first few Fermat numbers F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257, F 4 = 655373, n is odd and 8 if n is even.In fact, this is indeed true for all n : Theorem 3.1.Let n F be the th n Fermat number.Then,

Table 2
to compute digital roots of powers of large numbers.For example, Proposition 1.5.Digital root of an even perfect number (except 6) is 1.Proof.Every even perfect number m is of the form

Table 1 .
The multiplication table for digital roots is the familiar modulo 9 multiplication table with 0 replaced by 9.

Table 2 .
Digital roots of the powers of a natural number x form a cyclical sequence.