Slide 1Chapter 5: Number TheorySlide 3Selected Topics from Number TheoryPerfect NumbersDeficient and Abundant NumbersExample: Classifying a NumberAmicable (Friendly) NumbersGoldbach’s Conjecture (Not Proved)Example: Expressing Numbers as Sums of PrimesTwin PrimesTwin Primes Conjecture (Not Proved)Fermat’s Last TheoremExample: Using a Theorem Proved by FermatChapter 5Number Theory © 2008 Pearson Addison-Wesley.All rights reserved© 2008 Pearson Addison-Wesley. All rights reserved5-2-2Chapter 5: Number Theory5.1 Prime and Composite Numbers 5.2 Selected Topics From Number Theory5.3 Greatest Common Factor and Least Common Multiple 5.4 The Fibonacci Sequence and the Golden Ratio© 2008 Pearson Addison-Wesley. All rights reserved5-2-3Chapter 1Section 5-2Selected Topics from Number Theory© 2008 Pearson Addison-Wesley. All rights reserved5-2-4Selected Topics from Number Theory•Perfect Numbers•Deficient and Abundant Numbers•Amicable (Friendly) Numbers•Goldbach’s Conjecture•Twin Primes•Fermat’s Last Theorem© 2008 Pearson Addison-Wesley. All rights reserved5-2-5Perfect NumbersA natural number is said to be perfect if it is equal to the sum of its proper divisors.6 is perfect because 6 = 1 + 2 + 38 is not because 8 1 + 2 + 4.�© 2008 Pearson Addison-Wesley. All rights reserved5-2-6Deficient and Abundant NumbersA 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.© 2008 Pearson Addison-Wesley. All rights reserved5-2-7Example: Classifying a NumberDecide whether 12 is perfect, abundant, or deficient. SolutionThe proper divisors of 12 are 1, 2, 3, 4, and 6. Their sum is 16. Because 16 > 12, the number 12 is abundant.© 2008 Pearson Addison-Wesley. All rights reserved5-2-8Amicable (Friendly) NumbersThe 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.© 2008 Pearson Addison-Wesley. All rights reserved5-2-9Goldbach’s Conjecture (Not Proved)Every even number greater than 2 can be written as the sum of two prime numbers.© 2008 Pearson Addison-Wesley. All rights reserved5-2-10Example: Expressing Numbers as Sums of PrimesWrite each even number as the sum of two primes.a) 12 b) 40Solutiona) 12 = 5 + 7b) 40 = 17 + 23© 2008 Pearson Addison-Wesley. All rights reserved5-2-11Twin PrimesTwin primes are prime numbers that differ by 2.Examples: 3 and 5, 11 and 13.© 2008 Pearson Addison-Wesley. All rights reserved5-2-12Twin Primes Conjecture (Not Proved)There are infinitely many pairs of twin primes.© 2008 Pearson Addison-Wesley. All rights reserved5-2-13Fermat’s Last TheoremFor any natural number n 3, there are no triples (a, b, c) that satisfy the equation�.n n na b c+ =© 2008 Pearson Addison-Wesley. All rights reserved5-2-14Example: Using a Theorem Proved by FermatEvery 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.Solution2 27 16 9 4 3= - =