Ashley Majzun
Texas A&M University
Math 482 Final Report
Instructor: Dr. David Larson
May 7, 2013
Arithmetic Functions: The Euler
Function and The Sum of Positive Divisors
Function
Number Theory has been studied throughout the centuries. In southern Italy between
572-500 B.C., Pythagoras founded the famous Pythagorean School, a fraternity dedicated to the
study of mathematics, philosophy, and natural science. It was known as the brotherhood of the
Pythagoreans, and they dominated the first three centuries of classical Greek mathematics.
Pythagoreans believed that whole numbers completely described the universe and certain
numbers had mystical properties, thus their beliefs motivated the intense study of arithmetic and
number theory. For example: Perfect Numbers. For in the positive integers, is said to be a
perfect number is
. The mystical properties of numbers really fell into other categories
other than just mathematics. Saint Augustine once wrote, “Six is a number perfect in itself, and
not because God created all things in six days; rather, the converse is true. God created all things
in six days because the number is perfect.”
I. Basic Facts
Definition 0: Let
with and not zero. The greatest common divisor of
denoted
, is the greatest positive integer such that
and
. If
are said to be relatively prime.
Examples: (i) Let
and
(ii) Let
. Then
and
and ,
, then
and
. Then
So then
and
are relatively prime.
Theorem (Fundamental Theorem of Arithmetic): Every integer greater than 1 can be
expressed in the form
with
distinct prime numbers and
positive integers. This form is said to be the prime factorization of the integer. This prime
factorization is unique except for the arrangement of the
.
Proof: Assume, by way of contradiction, that is an integer greater than 1 that does not
have an expression as in the statement of the theorem. Without loss of generality, we may
assume that is the least such integer. Now cannot be a prime because it would then be of the
desired form. So is composite and
with
and
. But then and
are of the desired form due to minimality of ,from which it follows that is of the desired form,
a contradiction. So every integer greater than 1 has an expression of the desired form. We still
must show the uniqueness of such an expression. Assume that has two such expressions, say
With
distinct prime numbers;
distinct prime numbers; and
positive integers. Without loss of generality, we may assume that
and
. We must show that
and
Now, given a , we have
, which implies that
some . Similarly, given a , we have
for some . So
and ’s, we have
,
. Consequently,
Now assume, by the way of contradiction, that
may assume that
. Then
for some . So
for
; by the ordering of the ’s
for some .Without loss of generality, we
which implies that
Since
, we have that
from which
for some
. This is a contradiction. So
,
.
II. Arithmetic Functions
Definition 1: An arithmetic function is a function whose domain is the set of positive integers
Definition 2: An arithmetic function is said to be multiplicative if
whenever and are relatively prime positive integers. An arithmetic function is said to be
completely multiplicative if
for all positive integers and .
Theorem 1: Let
be an arithmetic function, and for
If
is multiplicative, then
with
, let
is multiplicative.
Proof: Let and be relatively prime positive integers. To prove that
multiplicative, we must show that
. We have
is
Since
,
we have
, each divisor
, and
of
can be written uniquely as
where
and each such product
corresponds to a divisor
of
,
. So
since f is multiplicative
as desired.
Example: Let
and
.
Then,
.
III. The Euler Phi-Function
Definition 3: Let
defined by
with
. The Euler phi-function, denoted
, is the function
Examples:
(i)
(ii)
since there are four positive integers less than or equal to 12 that are
relatively prime to 12: 1, 5, 7, and 11.
since there are twelve positive integers less than or equal to 13 that are
relatively prime to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12
(iii)
since there are four positive integers less than or equal to 5 that are
relatively prime to 5: 1, 2, 3, and 4.
Note that if
i.e.
is a prime number, then all positive integers less than
Theorem 2: The Euler phi-function
Proof: Let
are relatively prime to ,
is multiplicative.
and be relatively prime positive integers. We must show that
. Display the positive integers not exceeding
in matrix form as follows:
1
2
Consider the th row of the matrix above. If
, then no entry of the th row is
relatively prime to and so not relatively prime to
. [ Every entry of the th row is of the
form
where is an integer; if
then
and
so that
]
Hence, to determine how many integers in the matrix are relatively prime to
[which is
], it suffices to examine the th row of the matrix if and only if
; there are
such rows. Now, given such an th row, we must determine how many integers in the row
are relatively prime to
. The entries of such a row are
.
Since
, each of these integers is relatively prime to . Furthermore, these integers
form a complete system of residues modulo and so exactly
of these integers are relatively
prime to . Since these
integers are relatively prime to , they are relatively prime to
.
In summary, then, exactly
rows in the matrix contain integers relatively prime to
.
Each such row contains exactly
integers relatively prime to
. So
, the total
number of integers in the matrix that are relatively prime to
, is
as desired.
Theorem 3: Let
be a prime number and let
with
. Then
Proof: The total number of positive integers not exceeding
is clearly . The positive
integers not exceeding
that are not relatively prime to
are precisely the multiples of
given by
. There are clearly
such multiples. S the total number of
positive integers not exceeding
that are relatively prime to
is the difference of the above
quantities, and
, as desired.
Theorem 4: Let
with
. Then
Proof: Assume that
and that
numbers and
positive integers. Then,
with
distinct prime
Example: Let
, so the distinct prime numbers dividing 504 are 2, 3, and 7. So,
So there are exactly 144 positive integers not exceeding 504 that are relatively prime to 504.
IV. The Sum of Positive Divisors Function
Definition 4: Let
function defined by
with
. The sum of positive divisors function, denoted
Theorem 5: The sum of positive divisors function
Proof: The proof that
Theorem 6: Let
is multiplicative.
is multiplicative follows from Theorem 1.
be a prime number and let
with
. Then,
Proof: The positive divisors of
are precisely
divisors is
from the formula for the sum of the first
geometric series with ratio .
Theorem 7: Let
nonnegative integers. Then,
, is the
with
; the sum of these
terms of a
distinct prime numbers and
Proof: Let
with
nonnegative integers. Then,
Example: Let
distinct prime numbers and
. So
So there sum of the positive divisors of 504 is 1560.
X. Open Problems
This problem was taken out of the Elementary Number Theory textbook and there are three main
components. This question still remains open.
Are there infinitely many pairs of positive integers
First and foremost, we prove that
for
immediately that there could be infinitely many pairs.
Proof: Let
and
and
such that
?
and
. This will illustrate
. Using Theorem 4 and Theorem 7,
The second and third components deal with the Twin Prime conjecture and the Infinitude of
Mersenne primes, which are stated below.
Conjecture 1 (Twin Prime Conjecture): There are infinitely many prime numbers
is also a prime number
for which
Conjecture 2 (Infinitude of Mersenne primes): There are infinitely many Mersenne primes.
Note that any prime number expressible in the form
Mersenne prime.
with
prime is said to be a
The textbook argues that the truth of conjecture 1 and conjecture 2 would affirmatively answer
the open question. Although the answer is still unclear, I conclude with an interesting
observation. Note that
, which holds true for and
being twin
primes. A similar approach can probably be used for the Mersenne primes.
In addition to this open problem, there are many additional open problems that involve
arithmetic functions. One example is the question of whether infinitely many perfect numbers
exist.
References:
Strayer, James. "Arithmetic Functions." In Elementary Number Theory . Long Grove: Waveland
Press, Inc. , 1994. 77-92.
Strayer, James. "The Fundamental Theorem of Arithmetic." In Elementary Number Theory.
Long Grove: Waveland Press, Inc., 1994. 26-28.
"perfect number (mathematics) -- Britannica Online Encyclopedia." Britannica Online
Encyclopedia. http://www.britannica.com/EBchecked/topic/451491/perfect-number (accessed
February 18, 2013).