MAT 5400 (Elementary Theory of Numbers)
Fall 2020, Section 001, Course Reference Number 13433
Professor Drucker

ASSIGNED PROBLEMS

(last updated 4 p.m., October 5, 2020)

TEXTBOOK: John J. Watkins,
Number Theory: A Historical Approach,
Princeton University Press, 2014.

ADVICE: Assigned problems are the main means by which you will learn the course material. Keep your problem solutions together—separate from your class notes, in a binder that allows you to insert and remove pages, and organize it by chapter number, section number, and problem number. Flag the problems that you can’t completely solve, so that you can ask about them in class or in office hours.
    Start the assignment for a chapter as soon as you have read it. Don’t expect to be able to do all the assigned problems right away. Let your mind “stew” over the troublesome ones for a period of hours or days and try again. You may find that you can improve some solutions and make headway on others that you couldn’t do initially. Solve extra problems when you have trouble with a topic.
    Problems marked with a star (asterisk) should be submitted so that they can be corrected and graded. I’ll post the due dates in Canvas and/or on this assigned problems page.
    You may discuss problems with other students in the class, and you can always talk to me if you get really stuck, but the write-ups you turn in should be entirely your own. (This means that you shouldn’t be taking notes while you discuss problems with other students—you should just be brainstorming.) Don’t do “research” to find solutions to problems in other texts, and don’t solve starred problems with the help of tutors or other people not in the class. The purpose of the problems is to develop your problem-solving skills and your ability to write correct, logical, readable solutions. Just as in any other subject, your write-ups should consist of complete sentences, not vertical lists of expressions or equations.
    Hints and/or solutions to most of the problems are given at the back of the book. Don’t look at solutions except as a last resort. Your goal should be to develop confidence in your problem-solving skills by finding ways to solve problems and check your work without looking at solutions.
    Because so many hints and solutions are provided by the author, I will sometimes add problems of my own for you to work on. I’ll use letters instead of numbers to distinguish them from the problems in the text—for example, 7A and 7B would be my first two problems based on Chapter 7.
    I will also add a few exercises based on Professor Burger's lectures. I’ll preface those with a b, so for example the exercise for Lecture Five will be denoted b5 and the second exercise for Lecture Six will be denoted b6.2.

WARNING: The assignments are tentative. It is likely that there will be changes—some due dates may change; some problems may gain or lose asterisks; some problems may be added or deleted. In particular, I may add more of my own problems. If we get beyond Chapter 8, I’ll add assigned problems for the extra material we cover.
Chapter 1. Number Theory Begins
1.2, 1.4*, 1.7, 1.10, 1.12, 1.15, 1.16, 1.18, 1A*, 1B*, 1C*. Due Tuesday, Sept. 8.
Problem 1A. Show that there are infinitely many primitive Pythagorean triples {x, y, z} satisfying z = x + 1. Then show that there are infinitely many primitive Pythagorean triples {x, y, z} satisfying z = y + 2. Give five examples of each type.
Problem 1B. a. What is the sum of the first n positive even integers? b. What is the sum of the first n positive even squares? c. What is the sum of the first n positive odd squares? Prove your answers. (You may use the results of Problems 1.10 and 1.12 if you’ve done them.)
Problem 1C. Write down the last seven digits of your student ID. Cross out any zeros in the number. Next cross out any even digits at the right-hand end of the number that remains. Now the last three digits will form an odd number n between 111 and 999. Find a primitive Pythagorean triple {x, y, z} with y = n. Then find the corresponding squares in arithmetic progression. Are there other primitive Pythagorean triples with y=n? If so, find them. If not, explain why there aren’t any. Hint: It will help to factor n. If n = ab, where a ≤ b, then a ≤ 31 since a² ≤ ab = n ≤ 999.
Chapter 2. Euclid
2.2, 2.4, 2.6, 2.8, 2.11, 2.12, 2.15*, 2.17, 2.18, 2,22*, 2.23*.
Chapter 3. Divisibility
3.1, 3.3, 3.4*, 3.5*, 3.9, 3.11, 3.14*, 3.16, 3.17*, 3.18*, 3.21*, 3.26*, 3.27, 3.28, 3.34*, 3.35, b5, b6.1*, b6.2*. Also read 3.23 and 3.24.
Referring to the lecture notes for Chapter 3, do each part of 3.26 using each of the three equivalent definitions of congruence.
Chapter 4. Diophantus
4.7*, 4.8*, 4.12, 4.15*, 4.16, 4.18, 4.19, 4.20.
Chapter 5. Fermat
5.2*, 5.3, 5.6, 5.9, 5.14, 5.15*, 5.16*, 5.23, 5.24, 5.27*, 5.28*, 5.30, 5.31*, 5.32, 5.33*, 5.44, 5.46*.
Chapter 6. Congruences
6.1, 6.2, 6.6* (Turn in a solution to the second version.), 6.14, 6.16, 6A*, 6B* (See statements below).
Problem 6A*. Find all solutions (if any) to the following linear congruences.
   a. 19x ≡ 31 (mod 40)   b. 18x ≡ 30 (mod 40)   c. 18x ≡ 29 (mod 40)   d. 61x ≡ 78 (mod 101)   e. 60x ≡ 78 (mod 102)
Problem 6B*. Use the Chinese Remainder Theorem to find strong divisibility tests for 12 and 15. You can use the method in the proof or the other solution method.
Chapter 7. Euler and Lagrange
7.7*, 7.8, 7.9, 7.10*, 7.11, 7A, 7B*, 7.15, 7.16, 7.20*, 7.22, 7.23, 7.31, 7.34, 7.41*. Read 7.29, 7.30, 7.38, 7.40, 7.42, and 7.43.
Problem 7A. Find the smallest primitive root of 41.
Problem 7B*. Use the constructive method of Example 2 (following Proposition 7.4 of the lecture notes, which is the same as Problem 7.13) to find a primitive root of 71. This is a case where p − 1 is the product of three primes raised to the first power.
Chapter 8. Gauss
8.3*, 8.10*, 8.12, 8.18, 8.19, 8.20. Read the solutions to 8.4, 8.5, and 8.6.

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