Applications of the Complex Roots of Unity
Ghaith Hammouri
University of Hartford
In a differential equations class last year I saw an explicit function that generated a sequence of
alternating 0’s and 1’s which was used to select specific terms in a series. This sequence prompted me to
wonder what functions might produce other periodic sequences with longer strings of zeros. After
unsuccessfully searching for such functions in textbooks, I decided to discover my own. I used various
approaches until finally I found the answer in the complex roots of unity.
This paper starts by introducing a sequence of functions, {Gk}, formed using elementary complex analysis
that generates the desired string of 0's and 1's. Then we form two other functions, S and , based on the
{Gk}, to reveal information about prime decompositions of integers. Finally, we study the factorization
of Mersenne numbers and suggest a general formula for predicting periodic appearances of powers of
primes in their factorizations.
Introduction
Let {an} be a sequence of 0’s and 1’s. For any integer k, we say that {an} has period k if
1,
when k | n
0,
otherwise.
an =
For example, the sequence {0, 1, 0, 1, 0, 1,…} has period 2, and the sequence { 0, 0, 1, 0, 0, 1, …} has
period 3.
1 - The function Gk
Let Gk: N → {0, 1} be a sequence of period k. Our first goal is to find an expression that generates Gk.
For example, G2 could be defined by
G2(n) =
(-1) n  (1) n
.
2
(1)
This particular form is of interest because it provides insight into formulating sequences with larger
periods. Before trying to find a generator for Gk we note that the numbers 1 and -1 appearing in equation
(1) are the two complex square roots of unity. So a possible approach to formulate G3 would be to use the
_
1
3
1
3
i, w  - 
i , and 1.
three complex cube roots of unity: w  - 
2
2
2
2
We define G3 by
_
See Figure 1.
( w ) n  ( w ) n  (1) n
G3(n) =
.
3
1
1
3
- +
i
2 2
w
1
_
w
1
3
- +
i
2 2
Figure 1: The three cube roots of unity
_
These roots form an abelian group generated by 1, w, or w under complex multiplication. Consequently,
_
_
If 3|n, then wn = w n = 1. Otherwise, wn + w n = -1. Therefore, the function G3 will yield 1 whenever n is
a multiple of three and 0 otherwise.
The four complex roots of unity are 1, i, -1 and –i. The function G4 produces a sequence of period 4:
These four roots also form an abelian group.
G4(n) =
(i) n  (-i) n  (1) n  (-1) n
.
4
To extend these results recall that the kth roots of unity can be expressed as eiθ, where θ =
j = 1,2,…,k. Therefore, we have:
k
G k (n ) 
 (e
i(
2πj
and
k
2 jπ
)
n
k
i 1
k
)
.
For example,
G5 (n) =
(e
2 πi
n
5
e
4 πi
n
5
e
6 πi
n
5
5
e
8 πi
n
5
e
10 πi
n
5
)
.
We can use the identity eiθ = cos(θ) + i sin(θ) to rewrite G5 in trigonometric form as follows:
(cos
G5 (n) =
2πn
2πn
10 πn
10 πn
 i sin
)  ...  (cos
 i sin
)
5
5
5
5
5
2
(cos
=
(4) πn
(-2) πn
2πn
10 πn
2πn
4πn
 ...  cos
)  i(sin
 sin
 sin
 sin
 0)
5
5
5
5
5
5
.
5
Therefore,
5
G 5 (n ) 
2 jn π
)
5
.
5
 cos(
j1
In general,
k
G k (n ) 
2 jn π
)
k
.
k
 cos(
j1
The function Gk can be extended to a function defined on all real numbers. This extension is an even,
periodic function that takes on the values 0 and 1 on the integers.
For example,
G 5 (x) 
1
2
4
6
8
10
(cos( πx)  cos( πx)  cos( πx)  cos( πx)  cos( πx))
5
5
5
5
5
5
G 5 (2) 
1 2
2
2
4
 cos( 2π)  cos( 2π)  0.2 (0.3236 ...)  (0.1236 ...)  0 ,
5 5
5
5
5
G 5 (5) 
1 2
2
2
4
1 2 2
 cos( 5π)  cos( 5π)     1 .
5 5
5
5
5
5 5 5
Therefore,
and
The graphs of G4 and G5 are given in Fig.’s 2 and 3.
G4
Figure 2
G5
Figure 3
3
When k is even the roots of unity (except for z = 1 and z = -1) and their complex conjugates have the
same real part. Similarly, when k is odd the roots of unity (except for z = 1) and their complex conjugates
have the same real part. Consequently, we can reduce the required calculations rewriting the formulas for
Gk for the separate cases of k even and k odd.
For example, consider
2
4
6
8
10
(cos( πx )  cos( πx )  cos( πx )  cos( πx )  cos( πx ))
5
5
5
5
5
G 5 (x) 
5
2
4
4
2
(cos( πx )  cos( πx )  cos( πx )  cos( πx )  1)
5
5
5
5

5
2
4
(2 cos( πx )  2 cos( πx )  1)
5
5
.

5
In general, we conclude that
k 1
2
G odd (n ) 
1   cos(
j1
2 jn π
)
k
k
.
For the case when k is even we examine G6:
6
G 6 (n ) 
2 jn π
)
6
6
 cos(
j1
2nπ
4nπ
6nπ
8nπ
10nπ
12 nπ
)  cos(
)  cos(
)  cos(
)  cos(
)  cos(
)
6
6
6
6
6
6

6
nπ
2nπ
2nπ
nπ
cos( )  cos(
)  cos(nπ)  cos(
)  cos(
) 1
3
3
3
3
.

6
cos(
Therefore,
G 6 (n ) 
1  cos(nπ)  2 cos(
6
nπ
2nπ
)  2 cos(
)
3
3 .
In general,
k
2
G even (n ) 
1  cos(nπ)  2 cos(
j1
k
2 jπn
)
k
.
4
The function Gk can now be written in terms of Geven and Godd:
G k (n )  G 2 (n )G even (n )  G 2 (n  1)G odd (n )
Note that if n is even
G k (even )  G 2 (even )G even (n )  G 2 (odd )G odd (n )
 G even (n ) .
2 - The S function
Now let’s see how the Gk functions can be applied to the factorization of integers. Figure 4 gives a table
of values of Gk(n) for n and k = 1,...,12.
n
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
1
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
0
1
0
0
0
0
0
1
SUM
1
2
2
3
2
4
2
4
3
4
2
6
k
Figure 4
For each integer n the sum of numbers in the nth column is the number of divisors of n. This is expected
since the Gk(n) = 1 if and only if k|n, Note, in particular, that all prime values of n have exactly two
divisors.
Define S: Reals→ Reals by:
n
S(n )   G k (n ) .
k 1
From the discussion above we see that for all integers n S(n) is the number of divisors of n. Figure 4
shows the graph of S(n) for x = 10,…,23.
5
S(x):
10  x  23
Figure 5
We note in passing that the set of prime numbers can be characterized as the solution set of the equation
p
2   G k ( p) .
k 1
3 - Properties of S
Two properties of the S function follow directly from its definition.
1. For any prime number P and any positive integer n, S(Pn) = n + 1.
pn
By definition, S(Ρ n )   G k (Ρ n ) = the sum of the divisors of Pn = n + 1.
k 1
2. If  and  are relatively prime, S(  ) = S(  ) S(  ).
S(n) yields the number of integers that divide n. Since  and  are relatively prime, they have no
common nontrivial divisors. Therefore, the number of divisors of (  ) will be the product of the
number of divisors of  and  .
6
4 - Applications and Observations
The Divisor function1:  (n)
The divisor function  (n) , which is commonly found in number theory texts2, yields the sum of all the
divisors of an integer n. A slight modification of the S function enables us to write  (n) in terms of Gn.
n
σ(n )   kG k (n ) .
k 1
The following calculations illustrate the difference between  and S:
4
S(4)   G k (4)  G1 (4)  G 2 (4)  G 3 (4)  G 4 (4)  3.
k 1
4
σ(4)   kG k (4)  G1 (4)  2G 2 (4)  3G 3 (4)  4G 4 (4)  1  2  4  7.
k 1
Periodic Factorization
The Fundamental Theorem of Arithmetic guarantees that any integer, n, can be represented uniquely as a
product of primes. The Gk functions can be used to give a formula for the factorization of n.
Consider expression (2):


 G j (n)
p
n   p ij1
i
i 1
,
(2)
where the sequence p1, p2, p3, … represents the sequence of primes 2, 3, 5, ….
The function Gk tests whether an integer n contains any power of the prime pk. If so, 1 is added to the
exponent of that prime; if not, 0 is added.
For example, if n = 24,
24  (2
= 2
G 2 ( 24 )  G 4 ( 24 )  G 8 ( 24 ) ...
G 2 ( 24) G 4 ( 24) G8 ( 24)
)(3
G 3 ( 24 )  G 9 ( 24 )  G 27 ( 24 ) ...
)(5
G 5 ( 24 )  G 25 ( 24 )  G125( 24 ) ...

G j
j 1 Ρi
)...(p i
( 24 )
)
3G3 ( 24) 5.0...p i0
= 2 33 .
1
2
More on this function can be found at: http://mathworld.wolfram.com/DivisorFunction.html or
David M. Burton, "Elementary Number Theory," Allyn and Bacon, Inc.1976, p110 ff
7
5. Mersenne Numbers
Expression (2) can be modified to represent the factorization of numbers of the form 2 n – 1, which are
called Mersenne numbers. These numbers are of special interest when they yield a prime number in
which case they are called Mersenne primes.
Consider the following table of factored Mersenne numbers. Notice that 3 appears as a factor in every
second Mersenne number, 5 appears as a factor in every fourth Mersenne number, and 7 appears as a
factor in every third Mersenne number. The prime 2 does appear since all Mersenne numbers are odd.
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
2n-1
factored
1
(3)
(7)
(3) (5)
(31)
(3)2 (7)
(127)
(3) (5) (17)
(7) (73)
(3) (11) (31)
(23) (89)
(3) 2 (5) (7) (13)
(8191)
(3) (43) (127)
(7) (31) (151)
(3) (5) (17) (257)
(131071)
(3) 3 (7) (19) (73)
(524287)
(3) (5) 2 (11) (31) (41)
(7) 2 (127) (337)
(3) (23) (89) (683)
(47) (178481)
(3) 2 (5) (7) (13) (17) (241)
(31) (601) (1801)
(3) (8191) (2731)
(7) (73) (262657)
(3) (5) (29) (43) (113) (127)
(233) (1103) (2089)
(3) 2 (7) (11) (31) (151) (331)
Table 1: Factored forms of 2n – 1
The period of the prime pi is the smallest integer n such pi appears as a factor of the Mersenne numbers n,
2n, 3n, … We denote the period of pi by pi* .
For example, From Table 1 it appears that p3* = 2, p5* = 4, p7* = 3, and p11* = 10.
Theorem 1 provides a formula for pi*, its proof requires the use of Fermat’s Little Theorem.
8
Fermat's Little Theorem3 For any prime p and any integer a,
a)
ap  a (mod p), and
b)
If p does not divide a, there exists a smallest integer, d, such that ad - 1  0 (mod p) and d | p - 1.
Theorem 1: For any prime pi (> 2), there exists a unique integer M such that:
pi*  pi  1 ,
(3)
M
Proof: Using part b) of Fermat’s Little Theorem with a = 2 and p = pi, there is a smallest integer d such
that 2d - 1  0 (mod pi). That is, d is the smallest integer such that pi divides the Mersenne number 2d – 1.
Therefore, d = pi*. It follows that pi* | (pi – 1) from which expression (3) follows.
We conclude with two interesting conjectures on the factorization of Mersenne numbers. Each depends
up one the following assumption, which we believe to be true.
*
Assumption: From Theorem 1 we have that for every i > 2, 2 pi  Kp i  1 for some integer K. We
assume that GCD(pi ,K) = 1.
Conjecture 1: For any prime pi (i ≥ 2) and any positive integer n, let pin be a factor of a Mersenne
number. The period of pin is pi*pi(n-1).
Proof: (by induction): We first find the period of pi2.
*
Since pi* is the period of pi, there exists a unique integer K such that 2 pi  Kp i  1 . Therefore,
*
2 npi  (Kp i  1) n
n
  (Kpi ) j C(n, j) .
j 0
It follows that
n
2 npi  1   (Kp i ) j C(n, j)
*
j1
n
 (Kpi ) (Kpi ) j1 C(n, j)
j1
 (Kpi )[n  (Kpi )1 C(n,2)  ...  (Kpi )n2 C(n, n  1)  (Kpi )n1 C(n, n)] .
3
(4)
http://mathworld.wolfram.com/FermatsLittleTheorem.html
9
From our earlier assumption pi does not divide K. Hence, the smallest value of n for which we can factor
a second pi from the right-hand side of expression (4) is n = pi. In this case, pi2 is a factor of the right-hand
side. Therefore pipi* is by definition the period for pi2.
Assuming that the period of pin is pi*pi(n-1) we show that the period of pin+1 is pi*pin:
For some integer K,
p( n 1)p*i
2i
 (Kpin  1)
Therefore,
2
mpi( n 1) p*i
 (Kpni  1) m
m
=  (Kpin ) j C(m, j) .
j0
It follows that
2
mpi( n 1)p*i
m
 1   (Kpin ) j C(m, j)
j1
m
 (Kpin ) (Kpin ) j1 C(m, j)
j1
 (Kp )[m  (Kpin )1 C(m,2)  ...  (Kpin )pi 2 C(m, m  1)  (Kpin )pi 1 C(m, n)] . (5)
n
i
Once again, assuming pi does not divide K, the smallest value of m for which we can factor pi from the
right-hand side of expression (5) is m = pi. In this case, pin+1 is a factor of the right-hand side, therefore
pinpi* is by definition the period of pin+1.
One difference between factoring the Mersenne numbers {1, 5, 7, 15,…} and the sequence of integers
{1, 2, 3, 4,…} is the first appearance of a prime factor. For example, in the list of integers, 5 appears first
as a factor when n = 5 and then reappears as a factor every fifth term (5, 10, 15,…). The number 52 will
appear as a factor when n = (5)(5) and then reappear every (5)(5) terms (25, 50, 75,...). Furthermore, 53
will appear as a factor when n = (5)(5)2. And then reappear every (5)(5)2 times (125, 250, 375,...), and so
forth.
On the other hand, when factoring Mersenne numbers, 5 appears first as a factor when n = 4, and then
will reappear every fourth term (4, 8, 12,…). Furthermore, 52 will appear as a factor when n = (4)(5), and
then will reappear every (4)(5) terms (20, 40, 60,…). Continuing, 53 will appear as a factor when
n = (4)(5)2 and then will reappear every (4)(5)2 terms (100, 200, 300,…).
To see a parallel in these factoring processes we can rewrite expression (2) as follows:


G
n   pij0
j
pi pi
(n)
i 1
.
Replacing pi by pi* we can use a similar formula to factor the Mersenne numbers:


 G * j (n)
p p
2  1   pi
n
i2
j0
i i
.
(6)
10
We start the product at i = 2 since p1 = 2 can never be a factor of a Mersenne number.
This formula shows that the factoring of Mersenne numbers merely depends on the exponent (n). We are
making the assumption that the first time a prime factor appears it will have an exponent of 1, which is
the assumption is made earlier.
Assuming that expression (6) is true we can prove the following conjecture that is unsolved as far as we
know:
Conjecture 2: For any prime pk, the factors of ( 2
pk
1)
are square free.4
Proof: For any prime pk, expression (6) becomes

2 pk  1   p
i2

 G * j ( pk )
j 0 p i p i
i

Our task is to prove that
 G * j ( Ρk )
j0 p i p i
≤ 1 for all i > 2.

G
j 0
G * j (p k )
p p
i i
j ( pk )
p*ip
i
will only yield one if

G * 0 (p k )  G * 1 (p k )  G * 2 (p k ) 
p i pi
p i pi
p i pi
* j
p p
i i
... .
(7)
divides pk. Since pk is a prime, expression (7) will reduce to:

G
j 0
j ( pk )
p*ip
i
 G * (p k ) .
pi
(8)
*
Expression (8) shows that the sum reduces to a single term, which will be equal to one when p i
will be equal to zero otherwise. We conclude that pi can not be raised to a power higher than 1.
 p
k
, and
Finally, it might be useful to mention that such periodicity in prime factors is also observed for different
sequences (3n - 1, for example) though it becomes more complicated to denote.
Conclusion:
The Gk functions are selection functions that provide help in writing certain mathematical expressions,
especially those used by programmers, such as multi-statement functions. It is also important to mention
that there are a number of numeric functions that can easily be denoted using the Gk functions, for
example the Euler totient function. Although the Gk and S functions are not computational shortcuts, they
do provide insights that were a great help to enhance my understanding of prime numbers and complex
numbers.
4
Paulo Ribenboim, "The Book of Prime Number Records." Springer-Verlag. New York: 1988. Page 80.
11
References:
[1] http://mathworld.wolfram.com/
[2] David M. Burton, "Elementary Number Theory," Allyn and Bacon, Inc.1976
[3] Paulo Ribenboim, "The Book of Prime Number Records." Springer-Verlag. New York: 1988
[4] Niven & Zuckerman, "An Introduction to the Theory of Numbers." Wiley. New York: 1960
12