1/19 2/25 3/8 4/23 5/24 6/11 Total/110 % Please do not write in the spaces above.

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1 1/19 2/25 3/8 4/23 5/24 6/11 Total/110 % Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before attempting any problems. You must show all work, or risk losing credit. Be sure to answer all questions asked. To receive full credit on problems, they must not only be mathematically correct, but they must also be solved using the correct notation and terminology. The following list of all primes below 100 may or may not be helpful: 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 Good luck! MATH Version I Spring 2018 Dr. Morton Name: Exam II

2 1. (19 points) Answer the following questions to prove that there are infinitely many primes: a. What method of proof will be used here? b. There are two statements that can be true, involving the number of primes. What are these two statements? c. Can both of the statements in b be true at the same time? (Circle one): Yes No. d. Which one of the statements in part b do we start the proof by assuming it is true? e. If the statement in part d is true, what can you say about the primes? f. Based on the statement in part e, you can build a list. i. What is on the list (in words)? ii. What is on the list (in numbers, labelling variables as needed)? iii. What does it mean to not be on the list? g. Off of the enumerated list from f part ii, you can construct a special number X. Tell me the formula for X. h. Is it possible that X is on the list in f? (Circle one): Yes No. Explain. i. Based on your answer to part h, what must be true of the number X, in terms of divisors of X? What famous theorem tells you so? j. Now find the divisor(s) of X that you said must exist in part i. Are there any such divisors? Why or why not? k. We have arrived at an impossible situation. Which two parts of the proof above give rise to this impossible situation? l. What does part k tell us is therefore false? m. What must therefore be true?

3 2. (25 points) Dimensions: a. Fill in the following table of the k-faces of an n-cube. All entries must be filled in (even if the entry is a 0.) You do not have to show your work. Note: If a number is too large to enter into your calculator, write out what you would plug into your calculator. n - c u b e s k- faces b. How do you build a 12-dimensional cube by moving an existing 11-dimensional cube. Be precise in your answer. c. Give the coordinates of one vertex of a 5-dimensional cube. d. What is a tesseract?. 3. (8 points) a. What is a definition of pure mathematics? b. What is a definition of applied mathematics? c. What did G.H. Hardy s apology amount to (two different ideas)? 1. 2.

4 4. (23 points) The following list of all primes below 100 may or may not be helpful: 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 a. How many Mersenne primes are known, as of today? b. Could 2 "#" 1 be a Mersenne prime? Why or why not? c. Is 239 a Mersenne prime? Why or why not? Note: 239 is prime; you do not need to prove this part. d. Is 31 a Mersenne prime? Why or why not? e. It turns out that the 15 th Mersenne prime is 2 &'() 1. What is the corresponding perfect number? Note: All of these numbers are way too large to put in your calculator; instead tell me what you would need to type in a calculator if you had one that could handle such large numbers. Be careful with parentheses, notation, etc. f. Give two examples of twin primes. g. The number =2 ', 7 &/ How many divisors does this number have? Show all work, but do not find all of the divisors.

5 5. (24 points) The following list of all primes below 100 may or may not be helpful: 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 a. Use a factor tree to illustrate the Fundamental Theorem of Arithmetic on the number 650. b. Find all divisors of 650. c. Is 650 abundant, deficient, or perfect? Show all work. d. Is 899 prime? Use the fastest method, and show all work. Tell me the values of all divisors that must be checked here. e. Are 68 and 58 friendly numbers? To answer, i. First find all divisors of each of these numbers (you do not need to show work). Note: 68=2 ' 17 and 58=2 29. ii. Then for each of the two conditions needed for numbers to be friendly, show all work and tell me if the condition is met or not. If it is not met, tell me why it is not me. Condition 1: Condition 2: iii. Then tell me if 68 and 58 are friendly. (Circle one): Yes No.

6 6. (11 points) More Number theory: a. Complete the sieve of Eratosthenes process for 11. You do not have to write a list or anything. I have done the process for numbers below 11. You need to do the process for 11 finishing though the end of the table. Do not worry about the process for 13 or 17 or. (There should NOT be a lot of marks that you need to make here this should be a relatively quick question.) b. Are 55 and 88 relatively prime? Why or why not (be specific)? c. At the beginning of the proof that 2 is irrational, you write 2 in terms of two whole numbers, a and b. Tell me how to write 2 in terms of these numbers, and list any conditions that a and b might have.