SECTION 5-3 Selected Topics from Number Theory Slide 5-3-1.

Slides:

Advertisements

Similar presentations

Discrete Mathematics Lecture 3

Advertisements

Copyright © Cengage Learning. All rights reserved.

Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.

PrasadPrimes1 VEDIC MATHEMATICS : Primes T. K. Prasad

Prime Numbers A prime number is a number with exactly two factors The first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Chapter Primes and Greatest Common Divisors ‒Primes ‒Greatest common divisors and least common multiples 1.

Prime Numbers – True/False. 3. There are infinitely many primes. True We can prove this by assuming there aren’t: Multiply all the primes together,

CSE115/ENGR160 Discrete Mathematics 02/03/11 Ming-Hsuan Yang UC Merced 1.

Pythagorean Triples Big Idea:.

My special number Why I chose this number I chose this number because number 4 means death in china and I would have chosen 4 but I had to chose.

Number Theory – Introduction (1/22) Very general question: What is mathematics? Possible answer: The search for structure and patterns in the universe.

Date: Topic: Proofs of Triangle Congruence (6.3)

Homework Review notes Complete Worksheet #2. Homework State whether the conditional sentence is true or false 1.If 1 = 0, then 1 = – 1 True F F T.

Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.

My Favorite “Conjecture” By: none other than THE GREAT Ben Carroll “Goldbach’s Conjecture”

 2012 Pearson Education, Inc. Slide Chapter 5 Number Theory.

Section 5.1 Number Theory.

Example 1 Write the first 20 terms of the following sequence: 1, 4, 9, 16, … Perfect Squares These numbers are called the Perfect Squares. x

Warm-up Solve for missing variable: 1)(14√2)² = 14² + b² 2)A = (b1 + b2)h/2; for b1.

8.1 Pythagorean Theorem and Its Converse

Benchmark 40 I can find the missing side of a right triangle using the Pythagorean Theorem.

SECTION 5-1 Prime and Composite Numbers Slide

What are factors? What makes a number prime or composite?

Classifying Numbers Odd numbers like 1,3,5,7,9… Even numbers like 2,4,6,8,10… Prime numbers like 1,3,5,7,11… Composite numbers like 2,4,8,9,10… Square.

Section 8-1: The Pythagorean Theorem and its Converse.

© T Madas. A prime number or simply a prime, is a number with exactly two factors. These two factors are always the number 1 and the prime number itself.

* Students will Use the Converse of the Pythagorean Theorem to solve problems. * Use side lengths to classify triangles by their angle measures. * Why?

CONJECTURES. A conjecture is a statement that must be proved or disproved.

5 – 2: Solving Quadratic Equations by Factoring Objective: CA 8: Students solve and graph quadratic equations by factoring, completing the square, or using.

Geometry Section 7.2 Use the Converse of the Pythagorean Theorem.

Systems of Linear Equations in Two Variables. 1. Determine whether the given ordered pair is a solution of the system.

Pythagorean Theorem and Its Converse Chapter 8 Section 1.

Prime Numbers and Factoring. Which expression is equal to 4x6? (a) 2 × 2 x 2 (b) 2 × 3 × 4 (c) 4 × 5 × 3 (d) 6 × 6.

9.3 The Converse of the Pythagorean Theorem

Whiteboardmaths.com © 2004 All rights reserved

Section 8-3 The Converse of the Pythagorean Theorem.

9.3 The Converse of the Pythagorean Theorem

Exploring. Pythagorean Theorem For any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the.

 2012 Pearson Education, Inc. Slide Chapter 5 Number Theory.

Section 1.8. Proof by Cases Example: Let b = max{a, b} = a if a ≥ b, otherwise b = max{a, b} = b. Show that for all real numbers a, b, c

CONFIDENTIAL1 Today we will be learning about Prime Factorization. Warm-Up: 1.)3 x 5 x 7 = 2.)2 x 2 x 2 x 3 = 3.)2 x 3 x 3 x 5 = 4.)7 x 7 x 11 = GOOD AFTERNOON!

Chapter 4 With Question/Answer Animations 1. Chapter Summary Divisibility and Modular Arithmetic - Sec 4.1 – Lecture 16 Integer Representations and Algorithms.

Number Theory: Prime and Composite Numbers

Pretty Numbers Moloy De !. 6 6 is the smallest perfect number 1, 2 and 3 are the only proper positive divisors of 6 and = 6 6, 28, 496, and.

Perfect Numbers (3/26) Definition. A number is called perfect if it is equal to the sum of its proper divisors. Examples: 6 and 28 are the first two (check!).

Introduction to Chapter 4: Pythagorean Theorem and Its Converse

Warm Up Simplify the square roots

8.1 Pythagorean Theorem and Its Converse

Section 5.1 Number Theory.

9.3 The Converse of the Pythagorean Theorem

Sect. 9.2 & 9.3 The Pythagorean Theorem

The Converse Of The Pythagorean Theorem

Section 7.2 Pythagorean Theorem and its Converse Objective: Students will be able to use the Pythagorean Theorem and its Converse. Warm up Theorem 7-4.

Bellringer Simplify each expression 5 ∙ ∙ 8.

7.2 The Pythagorean Theorem and its Converse

Pythagorean Theorem and Its Converse

9.3 The Converse of the Pythagorean Theorem

Section 5.1 Number Theory.

Example 1 What is the area of a square with a side length of 4 inches? x inches? What is the side length of a square with an area of 25 in2? x in2? x.

9-2 Pythagorean Theorem.

Squares of Binomials Chapter 5 Section 5.6.

Section 5.1 Number Theory.

PROVING THE PYTHAGOREAN THEOREM

8.1 Pythagorean Theorem and Its Converse

9.2 The Pythagorean Theorem

Systems of Linear Equations in Two Variables

The Converse of the Pythagorean Theorem

CS 220: Discrete Structures and their Applications

8.1 Pythagorean Theorem and Its Converse

Chapter 1 Basics of Geometry.

Presentation transcript:

SECTION 5-3 Selected Topics from Number Theory Slide 5-3-1

SELECTED TOPICS FROM NUMBER THEORY Perfect Numbers Deficient and Abundant Numbers Amicable (Friendly) Numbers Goldbach’s Conjecture Twin Primes Fermat’s Last Theorem Slide 5-3-2

PERFECT NUMBERS Slide A natural number is said to be perfect if it is equal to the sum of its proper divisors. 6 is perfect because 6 =

DEFICIENT AND ABUNDANT NUMBERS Slide A natural number is deficient if it is greater than the sum of its proper divisors. It is abundant if it is less than the sum of its proper divisors.

EXAMPLE: CLASSIFYING A NUMBER Slide Decide whether 12 is perfect, abundant, or deficient. Solution The proper divisors of 12 are 1, 2, 3, 4, and 6. Their sum is 16. Because 16 > 12, the number 12 is abundant.

AMICABLE (FRIENDLY) NUMBERS Slide The natural numbers a and b are amicable, or friendly, if the sum of the proper divisors of a is b, and the sum of the proper divisors of b is a. The smallest pair of amicable numbers is 220 and 284.

GOLDBACH’S CONJECTURE (NOT PROVED) Slide Every even number greater than 2 can be written as the sum of two prime numbers.

EXAMPLE: EXPRESSING NUMBERS AS SUMS OF PRIMES Slide Write each even number as the sum of two primes. a) 12b) 40 Solution a) 12 = b) 40 =

TWIN PRIMES Slide Twin primes are prime numbers that differ by 2. Examples: 3 and 5, 11 and 13

TWIN PRIMES CONJECTURE (NOT PROVED) Slide There are infinitely many pairs of twin primes.

SEXY PRIME Sexy primes are prime numbers that differ from each other by six. For example, 5 and 11, 7 and 13, etc. Find another sexy prime… 11, 17 17,23 23, 29 Slide

FERMAT’S LAST THEOREM Slide For any natural number n 3, there are no triples (a, b, c) that satisfy the equation

EXAMPLE: USING A THEOREM PROVED BY FERMAT Slide Every odd prime can be expressed as the difference of two squares in one and only one way. Express 7 as the difference of two squares. Solution