UNIT I
DIVISIBILITY AND PRIME FACTORIZATION
Number theory or in older usage is called higher arithmetic, is a branch
of pure mathematics devoted primarily to the study of integers. German
mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the
queen of the sciences—and number theory is the queen of mathematics."
Concepts of divisibility and factorization which form basis and
foundation for the study of more advanced topics in number theory are
developed in this chapter. Main topics of this chapter are quite old, which
dates back 300 B.C. These concepts result to important properties and
theorems of integers, which are extremely useful in elementary number
theory.
a. Divisibility, division algorithm, greatest integer function
Divisibility-is a fundamental relation connecting one integer to another
- it is regarded to Euclid of Alexandria
- divisibility relation means “divides evenly with zero remainder”
- 2 divides 4; 4 does not divide 7
Definition 1: Let a,b∈Z. Then a divides b, denoted a|b, if there exists c∈Z such
that b=ac. If a|b, then a is said to be the divisor or factor of b.
Example 1: 2|4 since there exists c∈Z such that 4=2c where c=2.
4 does not divide 7 since there does not exist c∈Z such that 7=4c.
Propositions 1 and 2:
1. Let a,b,c∈ Z. If a|b and b|c, then a|c.
2. Let a,b,c,m,n ∈ Z. If c|a and c|b, then c|ma+nb.
Definition 2: The expression ma+nb is said to be an integral linear
combination of a and b.
Definition 3: Let x∈ R. The greatest integer function of x, denoted by [𝑥], is the
greatest integer less than or equal to x.
Example 2: a.) [5]=5, since the greatest integer less than or equal to 5 is
itself.
7
7
b.) [3]=2, since the greatest integer less than or equal to 3 is 2.
1
Lemma 1:
Let x∈ R. Then x-1< [𝑥]≤ x.
Theorem 1: (The Division Algorithm) Let a,b∈ Z with b>0. Then there exist
unique q,r ∈ Z such that a=bq + r, 0≤r<b.
𝑎
𝑎
Note: q=[𝑏] and r=a-b([𝑏])
Example 3: Find q and r as in the division algorithm if a=-7 and b=3.
−7
q=[ 3 ]=-3
r=-7-3(-3)=2
Definition 4: Let n∈Z. Then n is said to be even if 2|n and n is said to be odd if
2 does not divide n.
Tests for Divisibility
Divisibility Rules for 2,3,4,5,6,7,8,9, and 10 are as follows:
A number is divisible by 2 if the last digit is either 0, 2, 4, 6, or 8.
A number is divisible by 3 if the sum of the digits is divisible by 3.
A number is divisible by 4 if the last two digits is divisible by 4.
A number is divisible by 5 if the last digit is either 0 or 5.
A number is divisible by 6 if the number is divisible by both 2 and 3.
A number is divisible by 7 if the new number is divisible by 7 after
removing the last digit from the original number, doubling it, and subtracting
the result from the remaining digits. The procedure may be applied over and
over again as necessary.
A number is divisible by 8 if the last three digits is divisible by 8.
A number is divisible by 9 if the sum of the digits is divisible by 9.
A number is divisible by 10 if the last digit is 0.
2
Activity No. 1
Divisibility and Division Algorithm
Objectives:
divisibility;
1. To prove or disprove statements using the concept of
and
2. To find unique integers q and r guaranteed by the division
algorithm.
Direction: Accomplish the following problems. Show all pertinent solutions.
1. Prove or disprove each statement below.
a. 7|42
b. 4|70
c. 18|0
d. 14|175637
e. 17|998206
2. Find the unique integers q and r guaranteed by the division algorithm
with each dividend and divisor below.
a. a=46, b=6
b. a=297, b=14
c. a=-87, b=8
d. a=-480, b=27
3
b. Primes, Euclid's Theorem, Prime Number Theorem, Goldbach and Twin
Primes conjectures, Mersenne primes, Fermat primes
Prime Numbers
Definition 5: Let p ∈ Z with p>1. Then p is said to be prime if the only positive
divisors of p are 1 and p. If n ∈ Z, n>1, and n is not prime, then n is said to be
composite.
Lemma 2: Every integer greater than 1 has a prime divisor.
Theorem (Euclid) 2: There are infinitely many prime numbers.
Proposition 3: Let n be a composite number. Then n has a prime divisor p with
p≤√𝑛.
Sieve of Eratosthenes
By Proposition 3, the sieve of Eratosthenes is a method of finding all
prime numbers less than or equal to a specified integer n>1 by producing a
criterion satisfied by all composite numbers. If an integer greater than one
fails to satisfy this criterion, then the integer is a prime number. This
systematic procedure was named after Eratosthenes of Cyrene (276-194
B.C.)
Example 4: Suppose we want to find all prime numbers less than or equal to
50. Then by Proposition 3, we know that √50.≈7.07. The prime numbers less
than 7.07 are 2,3,5, and 7. Thus, from a list of the integers from 2 to 50, we
cross out all multiples of 2,3,5, and 7 except 2,3,5, and 7. All integers that
were crossed out are composite numbers while the remaining integers are
prime numbers.
11
21
31
41
2
12
22
32
42
3
13
23
33
43
4
14
24
34
44
5
15
25
35
45
6
16
26
36
46
7
17
27
37
47
8
18
28
38
48
9
19
29
39
49
10
20
30
40
50
Proposition 4: For any positive integer n, there are at least n consecutive
composite positive integers.
Conjecture (Twin Prime Conjecture) 1: There are infinitely many prime
numbers p for which p+2 is also a prime number.
4
Definition 6: Let x ∈ R with x>0. Then 𝜋(x) is the function defined by
𝜋(x)= |{𝑝:𝑝 𝑝𝑟𝑖𝑚𝑒;1<𝑝≤𝑥}|.
Note: The vertical bars denote cardinality of the set.
Example 5: a.) 𝜋 (20) = 8, since there are 8 prime numbers greater than 1
but less than or equal to 20.
b.) 𝜋 (45) = 14, since there are 14 prime numbers greater than
1 but less than or equal to 45.
Theorem (Prime Number Theorem) 3:
𝜋(𝑥)ln⁡(𝑥)
lim
= 1.
𝑥
𝑥→∞
The Prime Number Theorem says that, for large x, the quantity
𝜋(𝑥)ln⁡(𝑥)
𝑥
𝑥
is
close to or approaching 1. Also, the quantity ln 𝑥 is an estimate for 𝜋(𝑥) for
large x.
𝒙
𝐥𝐧 𝒙
𝝅(𝒙)
x
103
168
104
1229
105
9592
106
78498
107
664579
108
5761455
𝝅(𝒙)𝐥𝐧⁡(𝒙)
𝒙
Conjecture (Goldbach, 1742) 2: Every even integer greater than 2 can be
expressed as the sum of two (not necessarily distinct) prime numbers.
Example 6: a.) 5=2+3
b.) 6=3+3
5
Definition 7: Any prime number expressible in the form 2^ −1 with p prime is
said to be a Mersenne prime.
Example 7: a.) 3 is a Mersenne prime since 2^2 −1=3.
b.) 7 is a Mersenne prime since 2^3 −1=7.
Conjecture 3: There are infinitely many Mersenne primes.
Definition 8: Any prime number expressible in the form 2^(2^𝑛 )+1 with n𝜖Z
and n≥0 is said to be a Fermat prime.
Example 8: a.) 3 is a Fermat prime since 2^(2^0 )+1=3.
b.) 5 is a Fermat prime since 2^(2^1 )+1=5.
Conjecture 4: There are exactly five Fermat primes.
Conjecture 5: There are infinitely many prime numbers expressible in the form
𝑛^2+1 where n is a positive integer.
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Activity No. 2
Prime Numbers and Prime Number Theorem
Objectives:
1. To use the Sieve of Eratosthenes in finding prime numbers;
2. To find twin primes;
3. To find 𝜋(x); and
𝜋(𝑥)ln⁡(𝑥)
4. To compute
for a given x.
𝑥
Direction: Accomplish the following problems. Show all pertinent solutions.
1. Use the Sieve of Eratosthenes to find all prime numbers less
than 200.
2. Find all twin prime numbers less than 250.
3. Find
a. 𝜋(25)
b. 𝜋(110)
c. 𝜋(250)
𝜋(𝑥)ln⁡(𝑥)
4. Compute
for
𝑥
a. x=25
b. x=110
c. x=250
7
c. The greatest common divisor, Euclid's algorithm, integral linear
combinations
Greatest Common Divisors
Definition 8: Let a,b 𝜖Z with a and b not both zero. The greatest common
divisor of a and b, denoted (a,b), is the greatest positive integer d such that
d|a and d|b. If (a,b)=1, then a and b are said to be relatively prime.
Example 9: a. The divisors of 24 are …
The divisors of 36 are…
The greatest common divisor of 24 and 36 is 12, which is
denoted by (24,36)=12.
b. Every integer is a divisor of zero. Hence, (12,0)=12.
c. (12,35)=1
Proposition 5: Let a,b 𝜖Z with (a,b)=d. Then (a/d,b/d)=1.
Proposition 6: Let a,b 𝜖Z with a and b not both zero. Then
(a,b)=min{ma+nb: n 𝜖Z, ma+nb>0}
Note: The set-theoretic function min produces the minimum element of a set.
The greatest common divisor of two integers is the least positive number that
is expressible as an integral linear combination of the integers.
Definition 9: Let 𝑎1 , 𝑎2,…, 𝑎𝑛 𝜖Z with 𝑎1 , 𝑎2,…, 𝑎𝑛 not all zero. The greatest
common divisor of 𝑎1 , 𝑎2,…, 𝑎𝑛 denoted (𝑎1 , 𝑎2,…, 𝑎𝑛 ) is the greatest integer d
such that d|𝑎𝑖 = 1,2, … , 𝑛. If (𝑎1 , 𝑎2,…, 𝑎𝑛 ) =1, then 𝑎1 , 𝑎2,…, 𝑎𝑛 are said to be
relatively prime. If (𝑎𝑖 , 𝑎𝑗 ) =1 for all pairs i,j with i≠j, then 𝑎1 , 𝑎2,…, 𝑎𝑛 are said to
be pairwise relatively prime.
Example 10: (12,36,49) =1
* relatively prime but not pairwise relatively prime
Lemma 3: If a,b 𝜖Z , a≥b>0, and a=bq+r with q,r 𝜖Z , then (a,b)=(b,r).
8
Theorem (Euclidean Algorithm) 4: Let a,b 𝜖Z with a≥b>0. By the division
algorithm, there exist 𝑞1 , 𝑟1 𝜖Z such that
a=b𝑞1 + 𝑟1, 0≤ 𝑟1 <b.
If 𝑟1 >0, there exist (by the division algorithm) 𝑞2 , 𝑟2 𝜖Z such that
b=𝑟1 𝑞2 + 𝑟2, 0≤ 𝑟2 < 𝑟1 .
If 𝑟2 >0, there exist (by the division algorithm) 𝑞3 , 𝑟3 𝜖Z such that
𝑟1=𝑟2 𝑞3 + 𝑟3 , 0≤ 𝑟3 < 𝑟2 .
Continue this process. Then 𝑟𝑛 =0 for some n. If n>1, then (a,b)=𝑟𝑛−1. If n=1,
then (a,b)=b.
Example 11: Find (1180,482) by using the Euclidean Algorithm.
Let a=1180 and b=482. Then by the division algorithm, we have,
1180=482(2)+216.
Since 𝑟1=216>0, then by the division algorithm, we have,
482=216(2)+ 50.
Since 𝑟2 =50>0, then by the division algorithm, we have,
216=50(4)+16.
Since 𝑟3 =16>0, then by the division algorithm, we have,
50=16(3)+2.
Since 𝑟4 =2>0, then by division algorithm, we have,
16=2(8)+0.
Since 𝑟5 =0, then by the Euclidean algorithm, (1180,482)=⁡𝑟4 =2.
Example 12: Express (1180,482) as an integral linear combination of 1180
and 482.
From the Euclidean algorithm, (written generally in the form a=bq+r)
1180=482(2)+216
482=216(2)+ 50
216=50(4)+16
50=16(3)+2,
we rewrite the expressions in the form r=a-bq. Thus,
216=1180-482(2)
50=482-216(2)
16=216-50(4)
2=50-16(3).
Then using back substitution, we must come up with the integral linear
combination
2=1180(m)+482(n)
d. The least common multiple
Definition 10: Let a,b 𝜖Z with a,b>0. The least common multiple of a and b,
denoted [a,b], is the least positive integer m such that a|m and b|m.
9
e. The Fundamental Theorem of Arithmetic, Euclid's Lemma, canonical
prime factorization, divisibility, gcd, and lcm in terms of prime
factorizations
Lemma (Euclid) 5: Let a,b,p 𝜖Z with p prime. If p|ab, then p|a or p|b.
Corollary 1: Let 𝑎1 , 𝑎2,…, 𝑎𝑛 ,p 𝜖Z with p prime. If p|𝑎1 , 𝑎2,…, 𝑎𝑛 , then p|𝑎𝑖 for
some 𝑖.
Theorem (Fundamental Theorem of Arithmetic) 5: Every integer greater than
1 can be expressed in the form 𝑝1 𝑎1 𝑝2 𝑎2 …𝑝𝑛 𝑎𝑛 with 𝑝1, 𝑝2 ,…,𝑝𝑛 distinct prime
numbers and 𝑎1 , 𝑎2 ,…, 𝑎𝑛 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 𝑝𝑖 𝑎𝑖 .
Theorem 6: Let a,b 𝜖Z with a,b>0. Then (a,b)[a,b]=ab.
Corollary 2: Let a,b 𝜖Z with a,b>0. Then [a,b]=ab if and only if (a,b)=1.
Example 13. Find the prime factorization of 54.
54=2·27
=2·3·9
=2·3·3·3
=2·33
Example 14. Find the prime factorization of 888.
888=2·444
=2·2·222
=2·2·2·111
=2·2·2·3·37
=23 ·3·37
Example 15. Find [54,888] by using prime factorization.
From the prime factorizations of 54 in example 13 and 888 in example
14 written in canonical form, [54,888]= 23 ·33 ·37=7992.
Example 16. Find (54,888) by using prime factorization.
10
From the prime factorizations of 54 in example 13 and 888 in example
14 written in canonical form, (54,888)= 2·3=6.
f. Primes in arithmetic progressions, Dirichlet's Theorem on primes in
arithmetic progressions
Theorem (Dirichlet’s Theorem on Prime Numbers in Arithmetic Progressions)
7: Let a,b 𝜖Z with a,b>0 and (a,b)=1. Then the arithmetic progression
a,a+b,a+2b,…,a+nb,…
contains infinitely many prime numbers.
Theorem 7 was named after Peter Gustav Lejeune Dirichlet (18051859). He was born in Germany and a student of Carl Friedrich Gauss and
Gauss’s successor at Gottingen University.
Proposition 7: There are infinitely many prime numbers expressible in the
form 4n+3 where n is a nonnegative integer.
Lemma 6: Let a,b 𝜖Z. If a and b are expressible in the form 4n+1 where n is
an integer, then ab is also expressible in that form.
11
Activity No. 3
Prime Factorization, GCD and LCM
Objectives: 1. To find gcd using the Euclidean Algorithm and express integers
as an integral linear combination of the given integers; and
2. To find gcd and lcm using prime factorization.
Direction: Accomplish the following and show pertinent solutions.
1. Use the Euclidean Algorithm to find the greatest common
divisors below. Express each greatest common divisor as an integral
linear combination of the original integers.
a. (37,60)
b. (441,1155)
c. (793,3172)
2. Find the lcm and gcd of each pair of integers below using
prime factorization.
a. 100 and 105
b. 3780 and 8820
c. 423453 and 1484945
12
UNIT II
CONGRUENCES AND ITS ASSOCIATED THEOREMS
Congruences satisfy a number of important properties, and are
extremely useful in many areas of number theory. Carl Friedrich Gauss, a
German mathematician, developed the theory of congruences which is
considered as the greatest work ever in the theory of numbers.
Four theorems involving congruences will be presented in this chapter.
These theorems are the Chinese Remainder Theorem, Wilson’s Theorem,
Fermat’s Little Theorem, and Euler’s Theorem, which are extremely useful as
computational and theoretical tools in number theory.
a. Congruence, residue classes, complete residue systems, reduced
residue systems
Definition 1: Let a,b,m⁡𝜖 Z with m>0. Then a is said to be congruent to b
modulo m, denoted a≡b mod m, if m|a-b. If a≡b mod m, then m is said to be
the modulus of the congruence. The notation a≡ b mod m means that a is not
congruent to b modulo m; a is said to be incongruent to b modulo m.
Carl Friedrich Gauss was the greatest mathematician of the nineteenth
century and is generally regarded as one of the greatest mathematicians of all
time. He is best known for his Disquisitiones Arithmeticae, which is perhaps
the greatest work ever in the theory of numbers. In it, Gauss develops the
theories of congruences, and quadratic residues.
Example 1. 25≡1 mod 3, since 3|25-1.
Example 2. 18≡-3 mod 7, since 7|18-(-3).
Example 3. -9≡ 2 mod 5, since 5| -9-2.
Proposition 1: Congruence modulo m is an equivalence relation on Z.
Properties:
a. Let a𝜖 Z Since any integer divides 0, then m|0, from which
m|a-a and a≡a mod m. (Reflexive Property)
b. Let a,b𝜖 Z and assume that a≡b mod m. Then m|a-b, and we
have m|(-1)(a-b), or equivalently, m|b-a. (Symmetric Property)
c. Let a,b,c𝜖 Z and assume that a≡b mod m and b≡c mod m.
Then m|a-b and m|a-c. (Transitive Property)
Consequence 1: Z is partitioned into equivalence classes under congruence
modulo m.
13
Example 4. The equivalence classes of Z under congruence modulo 3 are
[0]
={x 𝜖Z:x≡0 mod 3}
={x 𝜖Z:3|x-0}
={x 𝜖Z:3|x}
={x:x=3n for some n 𝜖Z }
={…,-6,-3,0,3,6,…}
[1]
={x 𝜖Z:x≡1 mod 3}
={x 𝜖Z:3|x-1}
={x:x-1=3n for some n 𝜖Z }
={x:x=3n+1 for some n 𝜖Z }
={…,-5,-2,1,4,7,…}
[2]
={x 𝜖Z:x≡2 mod 3}
={x 𝜖Z:3|x-2}
={x:x-2=3n for some n 𝜖Z }
={x:x=3n+2 for some n 𝜖Z }
={…,-4,-1,2,5,8,…}
Consequently, Z is partitioned into the three equivalence classes
={…,-6,-3,0,3,6,…}
={…,-5,-2,1,4,7,…}
={…,-4,-1,2,5,8,…}
under congruence modulo 3.
Definition 2: A set of integers such that every integer is congruent modulo m
to exactly one integer of the set is said to be a complete residue system
modulo m.
Example 5: {0,1,2} is a complete residue system modulo 3.
Proposition 2: The set {0,1,2,3,…,m-1} is a complete residue system modulo
m.
Definition 3: The set {0,1,2,3,…,m-1} is said to be the set of least nonnegative
residues modulo m.
Proposition 3: Let a,b,c,d 𝜖Z such that a≡b mod m and c≡d mod m.
i. (a+c)=(b+d) mod m
ii. ac=bd mod m
14
Example 6: Addition and multiplication tables for the equivalence classes
under congruence modulo 3 on Z (may be written as 𝑍3 ) are as follows:
+
[0] [1] [2]
[0] [0] [1] [2]
[1] [1] [2] [0]
[2] [2] [0] [1]
·
[0]
[1]
[2]
[0]
[0]
[0]
[0]
[1]
[0]
[1]
[2]
[2]
[0]
[2]
[1]
Proposition 4: Let a,b,c,d 𝜖Z. Then ca≡cb mod m if and only if 𝑎 ≡b mod
(m/(c,m))
15
Activity No. 4
Congruences and Residue Systems
Objectives:
1. To prove or disprove statements involving congruences;
2. To find a complete residue system for a given modulo m; and
3. To construct addition and multiplication tables for equivalence
classes under congruence modulo m.
Direction: Accomplish the given problems and show pertinent solutions.
1. Prove or disprove each given statement.
a. 8≡5 mod 3
b. 10≡15 mod 4
c. -6≡ 4 mod 5
d. 0≡ -7 mod 7
e. -30≡19 mod 7
2. Find a complete residue system modulo 8 consisting entirely of
a. odd integers.
b. even integers.
3. Construct the addition and multiplication tables of the equivalence
classes under congruence modulo 8.
16
b. Linear congruences in one variable, Euclid's algorithm
Definition 4: Let a,b 𝜖Z. A congruence of the form 𝑎𝑥 ≡b mod m is said to be a
linear congruence in one variable (x).
Example 7. The congruence 2𝑥 ≡4 mod 3 is a linear congruence in one
variable x. Modulo 3 means that there are only three values that x can have,
namely, 0,1, and 2. Substituting these x values in the given linear congruence
2𝑥 ≡4 mod 3, shows that only x=2 results in a true congruence; hence the
solution set for this congruence is 2 as well as any integer congruent to 2
modulo 3, i.e. {…,-7,-4,-1,2,5,8,11,…}.
Theorem 1: Let 𝑎𝑥 ≡b mod m be a linear congruence in one variable and let
d=(a,m). If d| b, then the congruence has no solutions in Z. If d|b, then the
congruence has exactly d incongruent solutions modulo m in Z.
Porism 1: Let 𝑎𝑥 ≡b mod m be a linear congruence in one variable and let
d=(a,m). If d|b, the d incongruent solutions modulo m of the congruence are
given by
𝑚
𝑥0 +( 𝑑 )n,
n=0,1,2,…,d-1
where 𝑥0 is any particular solution of the congruence.
Example 8. Suppose we want to find all incongruent solutions of 16x≡8 mod
28. By Euclidean algorithm, we have
28=16(1)+12
16=12(1)+4
12=4(3)+0.
Thus, d=(a,m)=(16,28)=4. Since d|b, i.e. 4|8, then the congruence has four
incongruent solutions modulo 28. To find these four incongruent solutions, we
must first find 𝑥0 . Expressing (16,28) as a linear combination of 16 and 28, we
have
4=16(2)+28(-1).
The solvability of the linear congruence 16x≡ 8mod 28 for x is equivalent to
the solvability of the equation 16x-28y=8 for x and y.
Multiplying both sides of 4=16(2)+28(-1) by 2 we obtain
8=16(4)+28(-2).
Rewriting it in the form similar to 16x-28y=8 we have
16(4)-28(2)=8.
17
Hence, 𝑥0 =4 and 𝑦0 =2. Now, by Porism 1, all incongruent solutions of 16x≡8
mod 28 are given by
28
4+( 4 )n,
n=0,1,2,3
i.e. 4,11,18, and 25.
Definition 5: Any solution of the linear congruence in one variable 𝑎𝑥 ≡1 mod
m is said to be a multiplicative inverse of a modulo m.
Corollary 1: The linear congruence in one variable 𝑎𝑥 ≡1 mod m has a
solution if and only if (a.m)=1. If (a,m)=1, then 𝑎𝑥 ≡1 mod m has exactly one
incongruent modulo m.
Example 9. The inverse of 3 mod 11 is 4 since (4*3) mod 11=1.
c. System of linear congruences, Chinese Remainder Theorem
Theorem (Chinese Remainder Theorem) 2: Let 𝑚1 ,⁡𝑚2 ,…,⁡𝑚𝑛 be pairwise
relatively prime positive integers and let 𝑏1 ,⁡𝑏2 ,…,⁡𝑏𝑛 be any integers. Then the
system of linear conguences in one variable given by
x≡ 𝑏1 mod 𝑚1
x≡ 𝑏2 mod 𝑚2
.
.
.
x≡ 𝑏𝑛 mod 𝑚𝑛
has a unique solution modulo 𝑚1 𝑚2 ···𝑚𝑛 .
Example 9. Solve the system
x≡1 mod 3
x≡1 mod 4
x≡1 mod 5
x≡0 mod 7.
Solving for M, we have
M= 3*4*5*7=420
M1=4*5*7=140
M2=3*5*7=105
M3=3*4*7=84
M4=3*4*5=60.
18
Writing the preceding results in the form ax ≡ 1⁡mod m,
140≡1 mod 3
105≡1 mod 4
84≡1 mod 5
60≡1 mod 7.
Determining the residues, we obtain
2𝑥1 ≡1 mod 3
𝑥2 ≡1 mod 4
4𝑥3 ≡1 mod 5
2𝑥4 ≡1 mod 7.
Hence,
x=(1*140*2)+(1*105*1)(1*84*4)+(0*60*2)
=721
x≡301 mod 420.
d. Wilson's Theorem
Lemma 1: Let p be a prime number and let a𝜖Z. Then a has its own inverse
modulo p if and only if a≡ ±1 mod p.
Theorem (Wilson’s Theorem) 3: Let p be a prime number. Then
(p-1)!⁡≡-1 mod p.
The theorem was named after John Wilson, but he could not provide a proof.
It was first proved by Joseph Louis Lagrange in 1771. Lagrange is an Italian
mathematician and regarded as one of the greatest mathematicians in the
eighteenth century.
Proposition 5: Let n𝜖Z with n>1. If (n-1)! ≡-1 mod n, then n is a prime
number.
e. Fermat's Theorem, pseudoprimes and Carmichael numbers
Theorem (Fermat’s Little Theorem) 4: Let p be a prime number and let
a𝜖Z. If p| a, then
𝑎𝑝−1 ≡1 mod p.
19
Fermat’s Little Theorem is named for Pierre de Fermat (1601-1665).
He is a French mathematician and was the founding father of the modern
theory of numbers.
Corollary 2: Let p be a prime number and let a𝜖Z. If p| a, then 𝑎𝑝−2 is the
inverse of a modulo p.
Corollary 3: Let p be a prime number and let a𝜖Z. Then 𝑎𝑝 ≡a mod p.
Corollary 4: Let p be a prime number. Then 2𝑝 ≡2 mod p.
Definition 6: Let n be a composite integer. If 2𝑛 ≡2 mod n, then n is said to be
pseudoprime.
Example 10. 341 is a pseudoprime since by Fermat’s Little Theorem
210 = 1024 ≡ 1 mod 341.
Hence 2 340 ≡ 1 mod 341.
f. Euler's Theorem
Definition 7: Let n𝜖 Z with n>0. The Euler phi-function, denoted 𝜑(n), is the
function defined by
𝜑(n)=|{x𝜖 Z:1≤x≤n;(x,n)=1}|.
Example 11. 𝜑(10)=4 since there are four positive integers less than or equal
to 10 that are relatively prime to 10, namely, 1,3,7,and 9.
Theorem (Euler’s Theorem) 5: Let a,m𝜖 Z with m>0. If (a,m)=1, then
𝑎𝜑(𝑚) ≡ 1 mod m.
The Fermat’s Little Theorem was generalized by Leonhard Euler in
1760. Euler (1707-1783) was a Swiss mathematician who has the honor of
being the most prolific mathematician in history.
Definition 8: Let m be a positive integer. A set 𝜑(m) integers such that each
element of the set is relatively prime to m and no two elements of the set are
congruent modulo m is said to be a reduced residue system modulo m.
Example 12. The reduced residue system modulo 10 is given by the set
{1,3,7,9}.
20
Porism 2: Let m be a positive integer and let {r1,r2,…,r⁡𝜑(m)} be a reduced
residue system modulo m. If a is an integer with (a,m)=1, then the set {r1a
,r2a,…,r⁡𝜑(m)a} is a reduced residue system modulo m.
Corollary 5: Let a,m x𝜖 Z with m>0. If (a,m)=1, then 𝑎𝜑(𝑚)−1 is the inverse of a
modulo m.
21
Activity No. 5
Congruence Theorems
Objectives:
1. To solve system of linear congruences using the Chinese
Remainder Theorem;
2. To use Wilson’s Theorem in finding the least nonnegative
residue modulo m;
3. To prove statement using Fermat’s Little Theorem; and
4. To find the Euler phi-function of an integer; and
5. To find a reduced residue system modulo of an integer.
Direction: Accomplish the given problems and show pertinent solutions.
1. Solve the given system of congruences.
a.
x≡2 mod 6
x≡1 mod 7
x≡5 mod 8
b.
2x≡1 mod 3
3x≡2 mod 5
5x≡4 mod 7
2. Using Wilson’s Theorem, find the least nonnegative residue modulo m of
each integer n below.
a. n=30! ; m=31
b. n=88! , m=89
3. Prove that 910 ≡1 mod 11 using Fermat’s Little Theorem.
4. Find 𝜑(n) for all integral n between 1 and 25.
5. Find a reduced residue system modulo of
a. 15
b. 18
22
UNIT III
ARITHMETIC FUNCTIONS
Function is one of the extremely used concepts in mathematics and that
it plays a significant role in number theory. In Unit III we have already
encountered one example of such function, namely, the Euler phi-function. In
this chapter, we investigate the Euler phi-function and other related functions
more thoroughly.
a. Arithmetic functions, Multiplicativity
Definition 1: An arithmetic function is a function whose domain is the set of
positive integers.
Example 1. a.) The Euler phi-function 𝜑(10) is an arithmetic function.
b.) The number of positive divisors of an integer n is an
arithmetic function.
Definition 2: An arithmetic function f is said to be multiplicative if f(mn)=f(m)f(n)
whenever m and n are relatively prime positive integers. An arithmetic
function f is said to be completely multiplicative if f(mn)=f(m)f(n) for all positive
integers m and n.
Theorem 1: Let f be an arithmetic function and, for n𝜖Z with n>0, let
F(n)=∑𝑑|𝑛,𝑑>0 𝑓(𝑑).
If f is multiplicative, then F is multiplicative.
b. The Euler phi function
Theorem 2: The Euler phi-function 𝜑(n) is multiplicative.
Theorem 3: Let p be a prime number and let a𝜖Z with a>0. Then
𝜑(𝑝𝑎 ) = 𝑝𝑎 -𝑝𝑎−1 .
Theorem 4: Let n𝜖Z with n>0. Then
1
𝜑(n) = n ∏𝑝|𝑛,𝑝⁡𝑝𝑟𝑖𝑚𝑒 (1 − 𝑝)
where ∏𝑝|𝑛,𝑝⁡𝑝𝑟𝑖𝑚𝑒
For instance,
means “the product over all distinct prime divisors p of n”.
1
1
1
∏𝑝|10,𝑝⁡𝑝𝑟𝑖𝑚𝑒 (1 − )= (1 − ) (1 − ).
𝑝
2
5
Example 2. Determine 𝜑(888) using Theorem 4.
23
Since 888=23 ·3·37, then the distinct prime divisors of 888 are 2,3, and
37. Hence,
1
1
1
𝜑(888)= 888(1 − 2) (1 − 3) (1 − 37)=288.
This means that there are exactly 288 positive integers not exceeding 888
that are relatively prime to 888.
Theorem (Gauss) 5: Let n𝜖Z with n>0. Then
∑𝑑|𝑛,𝑑>0 𝜑(𝑑)= n.
Example 3. Show that ∑𝑑|10,𝑑>0 𝜑(𝑑)= 10.
We have
10=∑𝑑|10,𝑑>0 𝜑(𝑑)
10=⁡𝜑(10) + 𝜑(5) + 𝜑(2) + 𝜑(1)
=4+4+1+1
= 10
as desired.
24
Activity No. 6
Arithmetic Function and Euler phi- Function
Objectives:
1. To show that two integers are multiplicative; and
2. To find 𝜑(n) using Theorem 4.
Direction: Accomplish the following problems. Show all pertinent solutions.
1. Show that F(mn)=F(m)F(n) if m=3 and n=4.
2. Using Theorem 4, find 𝜑(n) for each given value.
a. 605
b. 1592
c. 4851
3. Prove that if n is a positive integer, then
𝜑(2n)=
𝜑(n),
if n is odd
2𝜑(n),
if n is even.
25
c. The number-of-divisors function and sum-of-divisors function
Definition 3: Let n𝜖Z with n>0. The number of positive divisors function,
denoted v(n), is the function defined by
v(n)= |{d𝜖Z:d>0;d|n}|.
Theorem 6: The number of positive divisors function v(n) is multiplicative.
Theorem 7: Let p be a prime number and let a𝜖Z with a≥0. Then
v(𝑝𝑎 )= a + 1.
𝑎
𝑎
𝑎
Theorem 8: Let n=𝑝1 1 𝑝2 2 ···𝑝𝑟 𝑟 with 𝑝1,⁡𝑝2 ,…,⁡𝑝𝑟 distinct prime numbers and
𝑎1 ,⁡𝑎2 ,…,⁡𝑎𝑟 nonnegative integers. Then
v(n)=∏𝑟𝑖=1(𝑎𝑖 + 1).
Example 4. Compute v(888) using Theorem 8.
We know that 888=23·3·37. The exponents of the prime factors of 888 are 3,1,
and 1. Hence,
v(888)= (3+1)(1+1)(1+1)=16.
This means that there are exactly 16 positive divisors of 888.
Definition 4: Let n𝜖Z with n>0. The sum of positive divisors function, denoted
𝜎(n), is the function defined by
𝜎(n)= ∑𝑑|𝑛,𝑑>0 𝑑
Theorem 9: The sum of the positive divisors function 𝜎(n) is multiplicative.
Theorem 10: Let p be a prime number and let a𝜖Z with a≥0. Then
𝑝𝑎+1 −1
𝜎(𝑝𝑎 )=
𝑎
𝑎
𝑝−1
𝑎
Theorem 11: Let n=𝑝1 1 𝑝2 2 ···𝑝𝑟 𝑟 with 𝑝1,⁡𝑝2 ,…,⁡𝑝𝑟 distinct prime numbers and
𝑎1 ,⁡𝑎2 ,…,⁡𝑎𝑟 nonnegative integers. Then
𝑎 +1
𝜎(n)=∏𝑟𝑖=1
𝑝𝑖 𝑖
−1
𝑝𝑖 −1
Example 5. Compute 𝜎(888) using Theorem 11.
Since 888=23·3·37, then
24 −1
32 −1
372 −1
𝜎(888)= ( 2−1 ) ( 3−1 ) ( 37−1 )=2280
which is the sum of the positive divisors of 888.
26
d. Perfect numbers
Definition 5: Let n𝜖Z with n>0. Then n is said to be a perfect number if
𝜎(n)=2n. Consequently, n is said to be a perfect number if 𝜎(n)-n=n.
Example 6. Since 𝜎(6)=12=2(6), then 6 is a perfect number. Equivalently, the
sum of all positive divisors of 6 other than 6, i.e. 1+2+3=6, is equal to 6. Note
that 6 is the first perfect number.
Theorem 12: Let n𝜖Z with n>0. Then n is an even perfect number if and only if
n=2𝑝−1 (2𝑝 − 1)
where 2𝑝−1 is a prime number (Mersenne prime).
Conjecture 1: There are infinitely many perfect numbers.
Conjecture 2: Every perfect number is even.
27
Activity No. 7
Number of divisors and sum of divisors functions and Perfect Numbers
Objectives:
1. To determine the number of divisors and sum of divisors of a
given number; and
2. To find perfect numbers.
Direction: Accomplish the given problems below. Show all pertinent solutions.
1. Compute v(n) for each value of n.
a. 605
b. 1592
c. 4851
2. Determine 𝜎(n) for each value of n.
a. 605
b. 1592
c. 4851
3. Find the next four perfect numbers after 28.
28
UNIT IV
QUADRATIC RESIDUES
Linear congruences in one variable was seen in Unit II. This chapter is
focused on the solvability of quadratic congruences in one variable. It will be
devoted on the special form of quadratic congruences and the Legendre
symbol which will culminated to the quadratic reciprocity law.
a. Quadratic residues and nonresidues
Definition 1: Let a,m𝜖Z with m>0 and (a,m)=1. Then a is said to be be a
quadratic residue modulo m if the quadratic congruence x2≡a mod m is
solvable in Z; otherwise, a is said to be a quadratic nonresidue modulo m.
Example 1. 1 is a quadratic residue modulo 11 since 12≡1 mod 11.
Proposition 1: Let p be an odd prime number and let a⁡𝜖Z with p| a. Then the
quadratic congruence x2≡a mod p has either no solutions or exactly two
incongruent solutions modulo p.
Porism 1: Let p be an odd prime number and let a⁡𝜖Z with p| a. If the
congruence x2≡a mod p is solvable, say with x=x0, then the two incongruent
solutions modulo p of this congruence are given precisely by x0 and p-x0.
Proposition 3: Let p be an odd prime number. Then there are exactly
incongruent quadratic residues modulo p and exactly
nonresidues modulo p.
𝑝−1
2
𝑝−1
2
incongruent
b. The Legendre symbol, Euler's Criterion, Gauss' Lemma
Definition 2: Let p be an odd prime number and let a⁡𝜖Z with p| a. The
𝑎
Legendre symbol, denoted by (𝑝), is
𝑎
(𝑝)=
1, if a is a quadratic residue modulo p
-1, if a is a quadratic nonresidue modulo p.
The Legendre symbol was named after Adrien- Marie Legendre (17521833), a French mathematician. Legendre is known principally for a work
entitled Elements de geometrie, a simplification and rearrangement of Euclid’s
Elements.
1
Example 2. a. (11)=1
29
2
b. (11)=-1
Theorem 1 (Euler’s Criterion): Let p be an odd prime number and let a⁡𝜖Z with
p| a. Then
𝑎
(𝑝)= 𝑎
𝑝−1
2
mod p
3
Example 3. (7) ≡27 mod 7≡-1 mod 7
Theorem 2: Let p be an odd prime number. Then
𝑝−1
2
−1
( 𝑝 )= −1
=
1, if p≡1 mod 4
-1, if p≡3 mod 4
Lemma (Gauss’s Lemma) 1: Let p be an odd prime number and let a⁡𝜖Z with
p| a. Let n be the number of least positive residues of the integers
𝑝−1
𝑝
a,2a,3a,…,⁡ 2 (a) that are greater than 2. Then
𝑎
(𝑝)= (−1)𝑛
c. The law of quadratic reciprocity
Theorem (Law of Quadratic Reciprocity) 3: Let p and q be distinct odd prime
numbers. Then
𝑝
𝑞
(𝑞 ) (𝑝) =−1(
=
𝑝−1 𝑞−1
)(
)
2
2
1, if p≡1 mod 4 or q≡1 mod 4 (or both)
-1, if 𝑝 ≡ 𝑞 ≡3 mod 4
The simplification of the law of quadratic reciprocity was introduced by Emil
Artin. He was born in Austria and was one of the leading mathematicians of
the twentieth century.
7
53
4
Example 4. (53)= ( 7 )= (7)=1
30
References
Strayer, James K. Elementary Number Theory, Lock Haven University:
Waveland Press, Inc., 1994, 2002
https://www.onlinemathlearning.com/divisibility-rules-6.html. Divisibility Rules
for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
https://en.wikipedia.org/wiki/Number_theory.Number Theory
https://www.math.uh.edu/~minru/number/hj01cs01.pdf. Summary: Divisibility
and Factorization
http://mathworld.wolfram.com/Congruence.html. Congruence
https://www.youtube.com/watch?v=hB34-GSDT3k. The Extended Euclidean
algorithm
http://gauss.math.luc.edu/greicius/Math201/Fall2012/Lectures/ChineseRemai
nderThm.article.pdf. The Chinese Remainder Theorem
https://www.youtube.com/watch?v=2-tdwLqyaKo. Using the Chinese
Remainder Theorem on a system of congruences
31
32
33
34
35
36
37