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1 Chapter 12: Prime Numbers Practice HW p. 88 # 1, 2, 6 Recall that a prime number p is a number whose only divisors are 1 and itself (1 and p). A number that is not prime is said to be composite. The following set represents the set of primes that are less than 100: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…} Here are some basic facts about primes. Theorem 1: Infinitely Many Primes Theorem. There are an infinite number of primes. Proof: 2 █ 3 Example 1: Starting with the prime p1 2 , use the proof of Theorem 13.1 to generate a list of several primes. Solution: █ 4 5 Fact: Since 2 is the only even prime, every odd prime is congruent to either 1 (mod 4) or 3 (mod 4), that is, if p is prime, then p 1 (mod 4) or p 3 (mod 4) . Some p 1 (mod 4) primes: 5, 13, 17, 29, 37, 41, 53, 61, … Some p 3 (mod 4) primes: 3, 7, 11, 19, 23, 31, 43, 47, … Theorem 2: Primes 3 (mod 4) Theorem. There are infinitely many primes congruent to 3 modulo 4. Proof: 6 █ 7 8 Example 2: Use the ideas of the Primes 3 (mod 4) Theorem to generate a list of primes congruent to 3 modulo 4. Solution: █ 9 Theorem 3: Primes 1 (mod 4) Theorem. There are infinitely many primes congruent to 1 modulo 4. Proof: Chapter 23 █ Fact: The techniques for proving the Primes 3 (Mod 4) Theorem does not work for the Primes 1 (Mod 4) Theorem. Example 3: Demonstrate a numerical example why the techniques for proving the Primes 3 (Mod 4) Theorem does not work for the Primes 1 (Mod 4) Theorem. Solution: █ Question? When do other families of a mod m have an infinite number of primes in its congruence class. The next theorem, which we state without proof, answers this question. 10 Theorem 4: Direchlet’s Theorem on Powers in Arithmetic Progressions. Let a and m be integers with gcd( a, m) 1 . Then there are infinitely many primes p where p a (mod m) Example 4: Are there infinitely many primes congruent to 9 (mod 11) ? To 22 (mod 33) ? Solution: █