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Name ______________________________________________ MAT 102 – A Survey of Contemporary Topics in Mathematics Professor Pestieau Take-Home Exam 1 – Number Theory Due Tuesday, February 16 INSTRUCTIONS Present your work neatly on this exam packet for all the problems below. Justify your answers to receive full credit. Problem 1 [10 pts] A primordial prime is a prime number of the form or product of all primes less than or equal to some prime p. , where For example, since , we can check that and both primordial primes. In fact, (29, 31) is a pair of primordial twin primes! Find the next pair of primordial twin primes after (29, 31). Justify your answer. is defined as the are Problem 2 [20 pts] After finishing a meal with your friend at the local Chinese restaurant, you open your fortune cookie and get the following “lucky” numbers: 18, 28, 51, 102, 315, 414 After staring at them for a few minutes (and scribbling on his napkin), your mathematicallyinclined friend tells you “That’s interesting… you have exactly 2 abundant, 2 deficient and 2 perfect numbers!” Is your friend correct? Justify your answer using the table below. Number Prime Factorization List of Proper Divisors Sum of Proper Divisors 18 28 51 102 315 414 Answer: _____________________________________________________ Deficient? Abundant? Perfect? Problem 3 [10 pts] Two distinct natural numbers are said to be amicable if the sum of the proper divisors of one of the numbers is equal to the other number, and vice-versa. For example, 220 and 284 are amicable numbers since the sum of the proper factors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and the sum of the proper factors of 284 is 1 + 2 + 4 + 71 +142 = 220. Show that 2620 and 2924 are amicable numbers using the table below. Number Prime Factorization Proper Divisors 2620 2924 Problem 4 [5 pts] Explain why every prime number is necessarily deficient. Sum of Proper Divisors Problem 5 [20 pts] In the 18th century, Goldbach conjectured in a letter to Euler that every natural greater than 5 is the sum of 3 primes. For example, and (note that these 3 primes do not need to be distinct). Show that this conjecture is true for the naturals 20 through 39 by completing the table below. Number Sum of 3 Primes Number Sum of 3 Primes Number 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Problem 6 Sum of 3 Primes [10 pts] In 1770, Joseph-Louis Lagrange proved that every natural number can be written as the sum of four integer squares (not necessarily distinct). So, for example, 3 = 12 + 12 + 12 + 02 and 310 = 172 + 42 + 22 + 12. Note that while 0 is an integer square, it is not a natural square. Write the numbers 7, 17, 177, and 1770 as the sum of four integer squares. Problem 7 [5 pts] A number of the form , where is prime, is called a Mersenne number – named after Marin Mersenne, a French monk of the 17th century. If for some prime the number is also prime, then it is called a Mersenne prime. Find the smallest Mersenne number that is composite. Problem 8 [5 pts] Construct a prime desert of length 2016 using the method shown in class (you can also see this construction in the PDF posted on the webpage). Leave your answer in terms of factorials. Problem 9 [15 pts] Note: You need to consult a list of prime numbers on the internet (or elsewhere) to answer the questions below. a) Identify the largest prime desert amongst the first 100 naturals. b) List all the prime deserts of length greater than 4 amongst the first 100 naturals. c) Identify the largest prime desert between the naturals 100 and 300.