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Chapter 5

Chapter 5. Number Theory. Chapter 5: Number Theory. 5.1 Prime and Composite Numbers 5.2 Large Prime Numbers 5.3 Selected Topics From Number Theory 5.4 Greatest Common Factor and Least Common Multiple 5.5 The Fibonacci Sequence and the Golden Ratio. Section 5-2.

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Chapter 5

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  1. Chapter 5 Number Theory 2012 Pearson Education, Inc.

  2. Chapter 5: Number Theory • 5.1 Prime and Composite Numbers • 5.2 Large Prime Numbers • 5.3 Selected Topics From Number Theory • 5.4 Greatest Common Factor and Least Common Multiple • 5.5 The Fibonacci Sequence and the Golden Ratio 2012 Pearson Education, Inc.

  3. Section 5-2 • Large Prime Numbers 2012 Pearson Education, Inc.

  4. The Infinitude of Primes There is no largest prime number. Euclid proved this around 300 B.C. 2012 Pearson Education, Inc.

  5. The Search for Large Primes Primes are the basis for modern cryptography systems, or secret codes. Mathematicians continue to search for larger and larger primes. The theory of prime numbers forms the basis of security systems for vast amounts of personal, industrial, and business data. 2012 Pearson Education, Inc.

  6. Mersenne Numbers and Mersenne Primes For n = 1, 2, 3, …, the Mersenne numbers are those generated by the formula 1. If n is composite, then Mn is composite. 2. If n is prime, then Mn may be prime or composite. The prime values of Mnare called Mersenne primes. 2012 Pearson Education, Inc.

  7. Example: Mersenne Numbers Find the Mersenne number for n = 5. Solution M 5 = 2 5 – 1 = 32 – 1 = 31 2012 Pearson Education, Inc.

  8. Fermat Numbers Fermat numbers are another attempt at generating prime numbers. The Fermat numbers are generated by the formula The first five Fermat numbers (through n = 4) are prime. 2012 Pearson Education, Inc.

  9. Euler’s and Escott’s Formulas for Finding Primes Euler’s prime number formula first fails at n = 41: Escott’s prime number formula first fails at n = 80: 2012 Pearson Education, Inc.

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