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Discrete Structure I
Number
Theory
FOR-IAN V. SANDOVAL
INTRODUCTION TO NUMBER THEORY
LEARNING OBJECTIVES
• Introduce Number Theory
• Discover how arithmetic started
• Recognize the applications of Number Theory
• Compare and contrast the types of numbers
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. .
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)
NUMBER THEORY
➢ It contains a list of "Pythagorean triples", i.e., integers such that.
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
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.
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.
NUMBER THEORY
Have you ever
thought about why 1
is “one”, 2 is “two”, 3
is “three”…..?
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.
NUMBER THEORY
NUMBER THEORY
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,…}
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.
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.
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
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
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
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
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.
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
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.

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Number Theory - Lesson 1 - 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.
  • 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.