Final project topic ideas for MATH 310 Fall 2015 (copyright Cassie Williams 2015)

These ideas are taken from many sources. In particular, several of these ideas and descriptions are taken from or inspired by a similar list by Kate Thompson (currently at Davidson College). All topics are flexible; not all subtopics must be explored, and tangents you find while researching may be followed (provided they don’t overlap too heavily with someone else’s topic). Including relevant historical information is encouraged. You are welcome to suggest your own project topic and we can discuss how to make it work! I have a few other ideas up my sleeve too, in case all of these get used. (Project guidelines are in a separate document.)

1. Rabid Rabbits: Gauss edition The purpose of this project is to explore the Gaussian Fibonacci

Gk and their comparison to the traditional Fibonacci numbers fk. First, a definition.

The kth Gaussian Fibonacci Gk is defined to be Gk = fk + ifk+1.

(a) List the first few terms of the Gaussian Fibonnaci sequence (remembering that f0 = 0).

Prove also that Gk = Gk−1 + Gk−2 for k ≥ 2. 2 2 (b) Define the norm N of a Gaussian x + iy to be N(x + iy) = x + y . Compute N(Gk) for the first few k. Do you see a pattern? Prove it. ! 1 1 (c) Let M = . Compute M n for various small powers of n. What pattern do you 1 0 2 n notice? Prove it. While you’re at it, prove that fn+1fn−1 − fn = (−1) . ! ! 1 + 2i 1 + i Gn+1 Gn (d) Let N = . Show that NM n−1 = . What does this mean 1 + i 1 Gn Gn−1 2 about Gn+1Gn−1 − Gn?

(e) Can you determine a formula for Gn+2Gn+1 − Gn+3Gn?  √ !k √ !!k 1 1 + 5 1 − 5 (f) If you haven’t seen it before, prove that fk = √  − 1 −  for 5 2 2 √ 1+ 5 k ≥ 1. Recall that 2 is also known as φ, which is also known as the golden ratio. Prove that lim fn+1 = φ. What is lim fn ? n→∞ fn n→∞ fn−k 2. The point is rational. In this problem, you will explore ways to find points with rational coordinates on circles (and other nice curves). In particular, we’ll find a connection between rational points on a unit circle and primitive Pythagorean triples.

• First, forget about the circle. – Using algebra (i.e., modular arithmetic) show that (x, y, z) is a Pythagorean triple if and only if for s > t x = s2 − t2, y = 2st, z = s2 + t2. – Recall that a triple is primitive if the three elements share no nontrivial factors. (So 32 + 42 = 52 is primitive, while 62 + 82 = 102 is not.) Using the parameterization above, generate a list of Pythagorean triples for various values 1 ≤ t < s ≤ 10. Conjecture and prove conditions on s and t such that (x, y, z) is a primitive triple. • Now it’s circle time. Start with X2 + Y 2 = 1. If you’re wondering why a unit circle (and you should), look at what would happen if you scaled the (3, 4, 5) or (5, 12, 13) triangles so that they were in/on the unit circle.

2 Start with a rational point on the unit circle (this means a pair (X,Y ) ∈ Q ). Say you start with (−1, 0). Consider straight lines with rational slope 0 ≤ m ≤ 1 going through (−1, 0). Look at where those lines intersect the unit circle (other than (−1, 0)). What do you see? What’s a second way of writing Pythagorean triples? Are these primitive? • Now let X2 +Y 2 = 2 and consider lines through the point (1, 1). Describe all rational points. • Why won’t this procedure work on X2 + Y 2 = 3? • Another instance where this connection between rational points and curves can be seen is in the study of elliptic curves. After, of course, looking up what an elliptic curve is, let’s consider the following curve: E : y2 = x3 + 17.

Clearly, this has a rational point P = (−2, 3). Let’s run through the same idea as before. Write the equation of a straight line with slope m going through P . Intersect it with E. How likely are you to get MORE rational points? (Hint: not very, but explain why.) How can we fix things? Well, we have a degree 3 polynomial now, so in effect we need more points. Thankfully, Q = (2, 5) is another rational point on E. Intersect the straight line going through P and Q with E. What do you see? How can you use THIS to now generate (potentially) infinitely-many points on E? If you complete this example and want to learn more, you need to start looking up the group law for elliptic curves, and if you really want a challenge, check out Mordell’s Theorem (1922).

2 3. Factoring is Hard...and That’s a Good Thing. It’s time for encryption (aka applied ). In particular, this project involves RSA.

• Give a (brief) historical outline of RSA. How does it differ from previously-used cryptosystems (hint: look up public key cryptography)? That is, what made it special? • Outline the algorithm. Why does it work? • Compute some examples (even if you choose small primes and short messages). • What are some of the well-known attacks on RSA? That is, what choices of p and q would be REALLY dumb? Could you explain, for example, Wiener’s attack? • When RSA was first made public in 1977, the authors made the following challenge. Let the RSA modulus be

n=1143816257578888676692357799761466120102182967212423625625618 42935706935245733897830597123563958705058989075147599290026879543541

and let e = 9007 be the encryption exponent. The ciphertext is

c=96869613754622061477140922254355882905759991124574 31987469512093081629822514570835693147662288398962 8013391990551829945157815154

What is the message? You may want to explore other RSA factoring challenges: http://en.wikipedia.org/wiki/ RSA_Factoring_Challenge

As a general resource for this project, here is a link to a very well-written survey of the pros/cons/whys/hows of RSA: http://crypto.stanford.edu/~dabo/papers/RSA-survey.pdf Last, but not least, here’s the original paper of Rivest, Shamir and Adleman: https://www.cs. drexel.edu/~jjohnson/sp03/cs300/lectures/p120-rivest.pdf

[Note: RSA has been adapted to use points on elliptic curves. Someone with interest in elliptic curves (and a semester of linear algebra) could investigate that cryptosystem as well.]

3 4. Why Calc May Be a Pre-Req to Number Theory. [Warning: This project makes use of MATH 237.] You have seen in calc 2 (or in any elementary discussion of sequences and series) that

∞ X 1 n2 n=1 converges (this is a p-series with p > 1 if nothing else). You may have even seen what it converges TO (regardless, you’re about to prove it). There are various proofs of this (the first was attributed to Euler in 1736). You’re going to do a calculus proof:

R 1 R 1 dxdy • The double integral 0 0 1−xy is an improper integral and could be defined as the limit of double integrals over the rectangle [0, t] × [0, t] as t → 1−. But if you expand the integrand as a geometric series, you can express the integral as the sum of an infinite series. Show that

∞ Z 1 Z 1 dxdy X 1 = 1 − xy n2 0 0 n=1

• Now you’re just going to evaluate that double integral! Start by making the change of variables u − v u + v x = √ and y = √ 2 2 This gives a rotation about the origin through the angle π/4. You may want to sketch the corresponding region in the uv-plane. Note that if, in evaluating the integral, you encounter either of the expressions 1 − sin θ cos θ or cos θ 1 + sin θ you may like the identity cos θ = sin((π/2) − θ) and the corresponding identity for sin θ. • What other proofs can you find (that you could also explain)? • In a number theory interpretation, you’ve just computed the exact value of ζ(2), where ζ represents the Riemann zeta function. This is very open-ended, but you should explore the history of the Riemann zeta function and its applications to number theory (if you want something concrete, look up Euler products and study the connection between Riemann zeta functions and counting primes). You could also investigate the Riemann hypothesis, a well-known and unsolved conjecture in number theory.

4 5. The Density of the Primes. We know there are infinitely primes, but we also know they are relatively rare among the infinitely many composite numbers. How rare are they? One way to phrase this question is to ask “What fraction of all are primes?” Here is a method of investigation for questions of this sort.

• Start with an easier question. Let E(x) = # {even integers n with 1 ≤ n ≤ x}, a function which counts the even numbers up to some bound x. (For example, E(3) = 1,E(4) = E(x) 2,E(5) = 2,....) Consider the ratio x , and compute it for several values of x. What E(x) number is E(x)/x always near? What is lim ? x→∞ x • The counting function for the primes is π(x) = # {p prime | p ≤ x}. Using a computer, generate values of π(x)/x for x at regular-ish intervals up to at least 5000. You should notice that π(x)/x is decreasing, and a reasonable question is “How fast does it decrease?” • Conjectured in about 1797, the following theorem answers this key question: π(x) The Theorem. For π(x) as above, lim = 1. x→∞ x/ ln x (In other words, when x is large, the number of primes less than x is approximately equal to x/ ln x.) Again using a computer, generate data which might support this theorem by π(x) computing, for various values of x up to at least 109, π(x), x/ ln x, and . x/ ln x • The Prime Number Theorem (PNT) was proved in 1896 by Jacques Hadamard and Ch. de la Vall´eePoussin (independently) using complex analysis, and “elementary” proofs (which did not rely on complex analysis but are by no means easy) were given in 1948 by Paul Erd¨os and Atle Selberg. Investigate and explain some of the history of the problem and the various who attempted to prove the PNT in the century between conjecture and first proof. If you’re feeling adventurous, look up and try to understand either of the proofs of the PNT. (If you’re interested in the complex analytic proof, ask me for some nice links.)  2 • Now consider S(x) = # y < x | y = m for some m ∈ Z , the counting function for squares in the integers. Using a computer, generate values of this function for various large x values. Then find a simple function of x that is approximately equal to S(x) for large x. (That is, conjecture a function f(x) such that S(x)/f(x) seems to have a limit of 1 as x → ∞.) It is ok to base your answer on the data you generate; bonus for you if you try to prove it! Z x dt • It turns out that π(x) is even closer to the value of than to x/ ln x. Use calculus to 2 ln t show that R x dt  lim 2 ln t = 1. x→∞ x/ ln x Then, using the series expansion Z dt (ln(t))2 (ln(t))3 (ln(t))4 = ln(ln(t)) + ln(t) + + + + ..., ln t 2 · 2! 3 · 3! 4 · 4! R x dt 9 compute (numerically on a computer) the value of 2 ln t for several x values up to 10 . Compare to your previous data; which is closer to π(x), the integral or x/ ln x?

5 6. Fermat’s Last Theorem. Fermat’s Last Theorem, conjectured in 1637 by Fermat in the margin of a book, states

“No three positive integers a, b, c satisfy an + bn = cn for integers n ≥ 3.”

The proof of this arrived in 1994 thanks to Andrew Wiles and uses beautiful (though quite deep and difficult) facts about elliptic curves and modular forms. Obviously, it is not an appropriate first-semester number theory project to cover THAT proof, but here are some things you may find interesting:

• The case of n = 4 was actually one of the few things in his life Fermat was known to have proved (or at least published)! It uses a technique called infinite descent which you see a lot in upper-level number theory and algebraic geometry courses. There’s a catch, though: Fermat showed there were no nontrivial solutions to a4 − b4 = c2. Your task: understand this proof (you may need to search for “Fermat’s right triangle theorem”), and explain how it relates to the n = 4 case of Fermat’s Last Theorem. • Infinite descent can also be applied to the case n = 3. This time though (shock of shocks), the first detailed proof is attributed to Euler. Work that one out too. • It’s not too bad to see that you only need to show Fermat’s Last Theorem fails for n = 4 and n = p for p an odd prime. Can you explain that? • Can you give a chronology of other proofs, mathematicians, and techniques that were used prior to Wiles’ final kill? • As an additional challenge, try this. The equation a2 + b2 = c2 obviously has many solutions in the positive integers, while a3 +b3 = c3 has none. Suppose we want to find positive integer solutions to a3 + b3 = c2. (1)

– One solution is (a, b, c) = (2, 2, 4). Find three other solutions in the positive integers by looking for solutions of the form (a, b, c) = (xz, yz, z2). (Not all choices of x, y, z will work, so you should find out which ones do.) 2 2 3 – Given one solution (a0, b0, c0) and n ∈ Z, show that (n A, n B, n C) is also a solution. Define a primitive solution to be one that does not have this form for any n ≥ 2 (i.e., no common factors in this arrangement). What are some examples of primitive solutions? Can you find some that, like (2, 2, 4), have a = b?

6 7. Striving for Perfection. A n is said to be perfect if

X d = n. 1≤d

For instance, 1 + 2 + 3 = 6 is a . So is 1 + 2 + 4 + 7 + 14 = 28.

• All perfect numbers that we can write down are even. In particular, they have a VERY special form. This result is actually due to Euler. Prove it:

“n is an even perfect number iff n = 2m−1(2m − 1) where m ≥ 2 and 2m − 1 is prime.”

Note: The numbers of the form 2m − 1 which are prime are called Mersenne primes. You may want to show (if you haven’t already) that 2m − 1 is prime implies m is prime. • Are there any odd perfect numbers? If we compute this sum of for various odd n, it seems like it is always less than n. If that were the case, we would have an easy proof that there are no odd perfect numbers! However, in fact this perceived pattern is false. Find an odd number n with the sum of its proper divisors larger than n. (This is why mathematicians insist on proof rather than relying on data!) • It is unknown if there are any odd perfect numbers. That doesn’t mean we don’t have some idea of what they’d have to look like (beyond being odd, that is): – Show that if n is an odd perfect number, then n = pam2 where p is an odd prime, p ≡ a ≡ 1 (mod 4), and m is an integer. – Show that if n = pam2 is an odd perfect number with p prime then n ≡ p (mod 8). • You may be curious about the alternatives to perfection? Well, you should look up abundant and deficient numbers. Some results you could then address: – Any proper of a deficient or perfect number is deficient. – Any multiple of an abundant or perfect number, other than the perfect number itself, is abundant. – There exist infinitely many even abundant numbers. There exist infinitely many odd abundant numbers. – If n = paqb where p and q are distinct odd primes and a and b are positive integers, then n is deficient. • What about a multiplicative analogue? Can you say anything about numbers n such that

Y d = n? 1≤d

7 8. The Seven Dwarfs of Number Theory. No, this is not about Disney. And there aren’t seven. But anyway.

The Greeks got very creative with numbers with certain properties (like perfect numbers) or pairs or sets of numbers with a property. Some of the ideas they coined are , betrothed numbers, friendly numbers, and lonely numbers. Amicable numbers can be regular or exotic. Some types of numbers that come from iterating some map on the integers, like sociable numbers, or happy numbers.

For this project, define and investigate several of these types of numbers, giving examples and proving any results about conditions for existence, infinitude, etc. This is very open-ended! Are there other such types of numbers?

8 9. Primes in Sequences. Even though prime numbers are relatively rare, they sometimes come in “bunches”. For example, the pair of primes 11 and 13 are only two apart, and the triple of primes 7, 13, and 19 are all 6 apart. In this project, you will explore primes that come in sequences, what is conjectured, and what is known.

• Prove that except when n = 2, n and n + 1 cannot both be prime. • There are many cases of primes which are two apart, like 11 and 13, or 29 and 31. These are called twin primes. Find some large-ish examples of twin primes (you could use a computer to search). How many twin primes are there less than 200? • What about triple primes? These are three primes that are each two apart, like with 3, 5, and 7. How many triple primes are there? Make a conjecture and prove it. • What about larger gaps? Look up cousin primes and sexy primes. What is known about these? • For each of these types of prime sequences, we can ask if they occur infinitely often. In the case of twin primes, we have a major conjecture which is still unproven:

The Twin Primes Conjecture: There are infinitely many prime numbers p such that p + 2 is also prime.

Unlike many open conjectures, very recent major progress has been made on this one! You might want to look up “primes in bounded gaps” to see what kind of progress I mean. In particular, while the actual papers are rather technical, you might want to read the abstracts and news releases about the paper by Zhang in 2013 and the follow-ups by Maynard and Tao in 2014. You could spend some time giving an overview of the problem, historical context, and discuss (generally) the tools that have been used to address the question. • Another kind of “primes in sequences” idea is a theorem of Dirichlet from 1837:

Primes in Arithmetic Progressions: Let a and m be integers with gcd(a, m) = 1. Then there are infinitely many primes that are congruent to a mod m.

We will show in class how to prove this for a = 3 and m = 4. Can you do it for a = 5 and

m = 6? (Consider numbers of the form 6p1p2 . . . pr + 5.) For what other pairs a and m can you prove the theorem? Give some history on Dirichlet and this theorem as well.

9 10. Approximating Irrationals. Diophantine equations are any polynomials with integer coeffi- cients which are to be solved in the integers or rational numbers. One application is to approxi- √ mating irrational numbers like 2 by rational numbers.

In this project, you should investigate rational approximation, including the history of the problem and who has worked on it. It is still an active area of research, and you could see what the “state of the art” is! You should also understand how these rational approximations are tied to Pell equations (which are Diophantine equations of the form x2 − Ny2 = 1). This material is in Chapter 9 of our textbook (and we won’t be able to cover it during class) but you should use outside resources as well. You could start by completing the results in Chapter 9, and then see what tangents you decide to investigate!

10 11. Testing, 2, 3, 5. Primes hold a key role in cryptography and coding theory, but deciding whether a given number is prime is a very difficult problem. Chapter 10 of our book investigates a variety of primality tests; that is, formulas or algorithms for finding or proving primes. You could cover that material, and then investigate other primality tests, like using elliptic curves. If you are so inclined, you could also play around with coding these primality tests and comparing the algorithms and their run times for primes of various sizes.

11