# Math 567 -- Elementary Number Theory

**MATH 567**

Elementary Number Theory

MWF 1:20-2:10, Van Vleck B119

**Professor:** Andrei Caldararu (andreic@math.wisc.edu)

*Office Hours:* Wednesdays 2:30-3:30, Van Vleck 605.

**Grader:** Yifan Peng, email: peng64@wisc.edu

*Office Hours:* TBA.

Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook Elementary Number Theory: Primes, Congruences, and Secrets, which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, it is available as a free legal .pdf. We will be using the (free, public-domain) mathematical software SAGE, developed largely by Stein, as an integral component of our coursework. There is a useful online tutorial. You can download SAGE to your own computer or use it online.

Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.

**Course Policies:** Homework will be due on Fridays. It can be turned in late only with *advance* permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.

Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for.

**Grading:** The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.

**Syllabus:**
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)

- Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)
- Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)
- Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)
- Feb 17-21: Public-key cryptography and RSA (3.1-3.4)
- Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers
- Mar 2-6: Quadratic reciprocity (4.1-4.4)
- Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)
- Mar 11:
*Midterm exam* - Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)
- Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem
- Apr 6-10: More diophantine equations, elliptic curves (6.1)
- Apr 13-17: Applications of elliptic curves (6.2-6.3)
- Apr 20-24: More applications of elliptic curves (6.4)
- Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?

**Video Lectures:**
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.

**Lecture for Mar. 23**: watch it here.**Lecture for Mar. 25**: watch it here.**Lecture for Mar. 27**: watch it here.**Lecture for Mar. 30**: watch it here.**Lecture for Apr. 1**: watch it here.**Lecture for Apr. 3**: watch it here.**Lecture for Apr. 6**: watch it here.**Lecture for Apr. 8**: watch it here.**Lecture for Apr. 10**: watch it here.**Lecture for Apr. 13**: watch it here.**Lecture for Apr. 15**: watch it here.**Lecture for Apr. 17**: watch it here.**Summary of the Apr. 20 lecture**: watch it here.

**Homework:**
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.

**Jan 31**: 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.

Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x.

a) Can you formulate a conjecture about the relationship between f(N) and N/log N?

b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)

c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.)

d) Give as good an upper bound as you can for f(N).

Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.

**Feb 7**: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like this to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19

**Feb 17**: 2.15, 2.18, 2.23, 2.26, 2.30.

Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.

**Feb 21**: Book problems: 3.4, 3.5, 3.6

Problem A. Prove that there are infinitely many primes p such that 2 is **not** a primitive root in Z/pZ. We break this up into steps.

Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.

Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?

Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)

Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.

**Feb 28**: Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.

**March 23**: 4.1 from the book.

Problem A. Give a prime factorization of the Gaussian integer 7+9i.

Problem B. Read the notes from here. Note that Z[i] satisfies a *reduction theorem*: if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d).

B.1. When d = 1+2i, show that, for each n in Z[i], there is a **unique** pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?

B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.

Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)

Now define phi(d) to be the number of elements s of S such that s and d are coprime.

C.1. Compute phi(1+2i) and phi(3).

C.2. Prove that the value of phi(d) does not depend on the choice of S.

C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)

**March 30**

Problem A. Express 50005 as the sum of two squares.

In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).

Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is *unbounded.*)

Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?

Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]

D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].

D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)
*Remark:* The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!

**April 17**

Book problems: 5.3,5.4,5.5

Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).

A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.

A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.

Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.

Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)

Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?

**April 24**

Book problems 6.1, 6.2, 6.5, 6.10.

Problem A.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class.

Problem A.2. The point P = (0,1) lies on each of these curves. For a =3 determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.

Problem B. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.