By Kraft, James S.; Washington, Lawrence C

IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique facts The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties special FactorizationPreliminaryRead more...

summary: IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique facts The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties specified FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties purposes of specific Factorization A Puzzle Irrationality

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**Example text**

More recently, Fn has been factored for 6 ≤ n ≤ 11, and many more have been proved to be composite, although they are yet to be factored. Many people now believe that Fn is never prime if n ≥ 5. Fermat primes occur in compass and straightedge constructions in geometry. Using only a compass and a straightedge, it is easy to make an equilateral triangle or a square. It’s a little harder to make a regular pentagon, but it’s possible. The constructions of equilateral triangles and regular pentagons can be combined to produce a regular 15-gon.

3 Euclid’s Original Proof 13 2 · 3 · 5 · 7 · 11 · · · pn . 4, p divides their difference, which is 1. This is a contradiction: p 1 because p > 1. This means that our initial assumption that there is a finite number of primes must be incorrect. Since mathematicians like to prove the same result using different methods, we’ll give several other proofs of this result throughout the book. As you’ll see, each new proof will employ a different idea in number theory, reflecting the fact that Euclid’s theorem is connected with many of its branches.

3. Find all positive divisors of the following integers: (a) 20 (b) 52 (c) 195 (d) 203 4. Find all positive divisors of the following integers: (a) 12 (b) 13 (c) 15 (c) 16 5. Prove or give a counterexample for the following statements: (a) If c | a and c | b, then c | ab. (b) If c | a and c | b, then c2 | ab. (c) If c a and c b, then c (a + b). 6. Recall that a number n is even if n = 2k and is odd if n = 2k + 1. Prove the following: (a) The sum of two even numbers is even. (b) The sum of two odd numbers is even.