Download MAT 102 – A Survey of Contemporary Topics in Mathematics

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MAT 102 – A Survey of Contemporary Topics in Mathematics
Professor Pestieau
Take-Home Exam 1 – Number Theory
Due Tuesday, February 16
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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.