This document provides an introduction to number theory, including: - Number theory is the study of integers and their properties - It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria - It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers - It discusses applications of number theory like public key cryptography and error-correcting codes

1. Discrete Structure I
Number
Theory
FOR-IAN V. SANDOVAL

2. INTRODUCTION TO NUMBER THEORY

3. LEARNING OBJECTIVES
• Introduce Number Theory
• Discover how arithmetic started
• Recognize the applications of Number Theory
• Compare and contrast the types of numbers

4. NUMBER THEORY
➢ Number theory or, in older usage, arithmetic is a branch of pure
mathematics devoted primarily to the study of the integers.
➢ It is sometimes called "The Queen of Mathematics“
➢ The word "arithmetic" is used by the general public to mean
"elementary calculations“ (+, -, *, /).
➢ It has also acquired other meanings in computer science, as
floating point arithmetic.
➢ Particularly in the study prime numbers as well the properties of
objects made out of the integers or defined as generalization of the
integers. .

5. NUMBER THEORY
➢ The first historical
find of an
arithmetical
nature is a
fragment of a
table: the broken
clay tablet
Plimpton 322
(Larsa,
Mesopotamia,
ca.1800 BCE)

6. NUMBER THEORY
➢ It contains a list of "Pythagorean triples", i.e., integers such that.

7. NUMBER THEORY
➢ Pythagorean mystics gave great importance to the odd and the
even.
➢ The discovery that √ 2 is irrational is credited to the early
Pythagoreans (pre-Theodorus )
➢ By revealing (in modern terms) that numbers could be irrational,
this discovery seems to have provoked the first foundational crisis
in mathematical history;
➢ Its proof sometimes credited to Hippasus
Hippasus

8. NUMBER THEORY
➢ Āryabhaṭa (476–550 CE) showed that pairs of
simultaneous agreement n ≡ a 1 mod m 1
could be solved by a method he called
pulveriser
➢ this is a procedure close to the Euclidean
algorithm, which was probably discovered
independently in India. Āryabhaṭa
➢ Āryabhaṭa seems to have had in mind
applications to astronomical calculations.

9. NUMBER THEORY
➢ lived in the third century, that is about 500
years after Euclid
Diophantus of Alexandria
➢ Six out of the thirteen books of
Diophantus's ”Arithmetica” survived in the
original Greek and four more books
survived in an Arabic translation
➢ ”Arithmetica” is a collection of worked-out
problems where the task is to find out
rational solutions to a system of polynomial
➢ equations or algebraic equations.

10. NUMBER THEORY
Have you ever
thought about why 1
is “one”, 2 is “two”, 3
is “three”…..?

11. NUMBER THEORY
➢ The numbers we write are made up of algorithms, (1, 2, 3, 4, etc)
called arabic algorithms, to distinguish them from the roman
algorithms (I; II; III; IV; etc.)
➢ The Arabs popularize these algorithms, but their origin goes back
to the Phenecian merchants that used them to count and do their
commercial countability.

12. NUMBER THEORY

13. NUMBER THEORY

14. TYPES OF NUMBERS
➢ Counting Numbers
- positive whole numbers excluding zero or {1,2,3,4, 5…} also
called natural numbers
➢ Whole Numbers
- positive integers including zero or {0,1,2,3,4, 5…}.
➢ Integers
- numbers formed by the natural numbers including 0 together with
the negatives of the non-zero natural numbers or {…,-3,-2,-
1,0,1,2,3,…}

15. TYPES OF NUMBERS
➢ Rational Numbers
- numbers that can be written as fraction and whose numerator and
denominators are integers provided that the denominator is not equal
to 0
- it can also be written in decimal form as terminating decimal or as an
infinite repeating decimal
- Some examples of rational numbers are
➢ Real Numbers
- numbers compromised all rational and irrational numbers
➢ Imaginary Numbers
- the square root of negative one
- Any real number times I is an imaginary number some examples are
i,4i, -6.3i.

16. TYPES OF NUMBERS
➢ Complex Numbers
- the combination of real numbers and imaginary number (non-real
numbers) some examples are
➢ Odd Numbers
- a number when divided by 2 contains a remainder of 1.
- Mathematically, n is odd if there are exist a number k, such that
n=2k+1 where k is an integer.
➢ Even Numbers
- a number divisible by 2
- Mathematically n is even if there exist a number k, such that
n=2k where k is integer.

17. TYPES OF NUMBERS
➢ Prime Numbers or A Prime
- a natural numbers greater than 1 that has no positive divisors other
than 1 and itself, some example are 2,3,5,11.
➢ Composite Numbers
- a positive integer which has a positive divisor other than 1 or itself
- in other words any positive integer greater than 1 that is not a prime
number
- some examples are 4, 6, 8, 9, 10, etc.
➢ Perfect Numbers
- a positive integer that is equal to the sum of its proper positive
divisors, that is, the sum of its positive divisors excluding the number
itself
- Some examples are 6, 28, 496, 8128, 33550336

18. TYPES OF NUMBERS
➢ In symbols
R – real numbers
Q – rational numbers
N – natural numbers or counting numbers
W – whole numbers
Z – integers
Z – positive integers
Z – negative integers

19. NUMBER THEORY APPLICATION
➢ Number theory can be used to find out some of the important
divisibility tests, whether a given integer n is divisible by an integer
m

20. PUBLIC KEY CRYPTOGRAPHY
➢ Everybody has a key that encrypts and a separate key that
decrypts
➢ They are not interchangeable!
➢ The encryption key is made public
➢ The decryption key is kept private
➢ Public key cryptography goals
- Key generation should be relatively easy
- Encryption should be easy
- Decryption should be easy (with the right key
- Cracking should be very hard

21. PUBLIC KEY CRYPTOGRAPHY
➢ Number Theory for Digital Cash
- The whole of encryption works due to number theory. As a
result, security of transactions is ensured.
- If it were not for number theory, your money will not be safe in
your bank, information about you could be accessed by anyone.
➢ Error-Correcting Code
- is an algorithm for expressing a sequence of numbers such that
any errors which are introduced can be detected and corrected
(within certain limitations) based on the remaining numbers.

22. PUBLIC KEY CRYPTOGRAPHY
➢ Encrypting and Decrypting RSA messages
- Formula is c = me mod n
➢ Quantum computers
- A quantum computer could (in principle) factor n in reasonable
time
• This would make RSA obsolete!
• Shown (in principle) by Peter Shor in 1993
• You would need a new (quantum) encryption algorithm to
encrypt your messages

23. References
• Arefin, S. (2016). Number Theory. Retrieved from https://www.slideshare.net/SamsilArefin2/number-theory-70169905
• Aslam, A. (2016). Discrete Mathematics and Its Application. Retrieved from https://www.slideshare.net/AdilAslam4/number-
theory-in-discrete-mathematics
• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science University
of Colorado.