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Module 5 Homework Name: email address: phone number: Who helped me: Who I helped: Homework rules: Front side only. Keep the questions and your answers in order. If you send it pdf, send it in a single scanned file. ([email protected]) If you turn it in personally, have the receptionist date stamp it and put it in my mailbox. (651 PGH – 8am to 5pm) 79 point assignment 1 5 points 1. Find two examples where for 4 distinct numbers a, b, c, and d: a 2 b2 2 c2 d 2 What tips can you provide for others on how to find the numbers? 2 5 points 2. Show with examples and discuss why it’s true: A. The product of any 3 consecutive natural numbers is divisible by 6. B. The product of any 4 consecutive natural numbers is divisible by 24. C. The product of any 5 consecutive natural numbers is divisible by 120. You must use 12 different numbers in the example. If you use 5, you may only use it in one example. Fact: The product of n consecutive integers is divisible by n! D. Why is this fact true? 3 10 points 3. Create numbers of the form: 22 3 n Calculate the numbers for n = 1, 2, 3, 4: For which n does the number exceed the limit on number gossip? Which of the following vocabulary words (from “number gossip”) apply to each number? even odd algebraic transcendental square-free deficient, abundant, perfect, prime happy unhappy 4 5 points 4. Analyze the following patter 1x1 11 x 11 111 x 111 1111 x 1111 (R 5 ) 2 = = = = = 1 121 12321 1234321 123454321 (R 9 ) 2 = ___________________________ The answers are consecutive integer palindromes. What does that sentence say in regular English? Write 2 brief paragraphs about why this works from R1 to R9. 5 5 points 5. Find a pair of twin primes other than the one in the example. Note that they are of the form (n 1, n + 1) with the composite number, n, between them. (example, 5 and 7, n = 6). Now show that 2n 2 2 can be represented as the sum of 2, 3, 4, and 5 perfect squares (not necessarily distinct summands here) example: representing 2(36) +2 = 74 as two perfect squares: 25 + 49. Pick a prime number > 25 and call it p? Does the same fact, this time with 2p 2 2 work like this? Show that it might or might not – trade this around with study buddies to find some of each. 6 6 points 6. Theorem: If n divides m, the Rn divides Rm. Where Rn is the repunit with n 1’s. Use this theorem to find factors of R10. Then factor R10 to primes. 7 6 points 7. Find a set of prime numbers whose difference is 3. Are there many of these or few? 8 6 points 8. A prime triplet is {p, p + 2, p + 6}. when p and p + 2 are twin primes. Give 3 examples of 3 prime triplets. Why does this formula work? hint: see primes and mod 6 from Module 2 9 6 points 9. Using the prediction formula for the number of primes fewer than a given x, x N= , find the predicted number of primes below 50. Now, using a list of ln x primes to count (easily available on the internet), how many are actually between 1 and 50? 10 8 points 10. Theorem: If p and p2 + 8 are prime, then so is p3 + 4. Find two examples of this theorem or one example and an explanation of why there’s not a second example. 11 3 points 11. Find a gap of length at least 7 between two primes using the way I showed you in class. What are the two primes? 12 4 points 12. Give an arithmetic sequence with a leading term a and a difference d that are relatively prime. Show that it has many primes. Give an arithmetic sequence with a first term a and a difference d that are not relatively prime. Show that it’s prime free. 13 10 points 13. A. B. List 5 ways in which lucky numbers and primes are alike. List 5 ways in which lucky numbers and primes are different. Use good grammar and spelling. Make sure a reasonable person can understand what you’re saying. 14