# Number Theory

You all know **Number Theory**, the branch of mathematics which deals with the properties of the natural numbers. What's not so well known about Number Theory, formerly History of Number Theory, is that it is the highest level Math course in the CTY program. It requires previous enrollment outside of CTY in Geometry and Algebra II. Its course code is THEO. Number Theory is only offered at Lancaster and is considered a quintessential Lancasterian course. It is quite popular with both students and staff and has developed many rich traditions.

## Contents

## Course Description

Number Theory, the subject, is the branch of mathematics which deals with the properties of the natural numbers. The material covered in the course is roughly equivalent to a semester of undergraduate-level number theory, as taught at a college or university. Students learn important results in topics such as integer factorization, continued fractions, modular arithmetic, and Gaussian integers. The presentation of these subjects is quite rigorous; very few stated results, if any, are left unproven. Additionally, students spend some of the last week researching independent projects on related topics, which they present during the final days of class. Although Number Theory lies well outside the traditional high school mathematics curriculum, its value is borne out by the fact that at least two past members of the United States team at the International Mathematical Olympiad have cited the course as a significant factor in their mathematical successes.

## Class History

Number Theory was started by Pomm and Timmer, two longtime math instructors at Lancaster, in 1991. Their reason for creating the new course was simple: they wanted an excuse to work together as co-instructors, and number theory seemed to be an ideal topic for a new math course. They realized early on that number theory, which is usually taught as an intermediate undergraduate course for math majors in college, would be a particularly challenging subject for younger students, even students of CTY's academic caliber. Thus, they sought ways to make the material accessible and entertaining, and it is safe to say that they succeeded. They peppered the curriculum with skits and jokes and other alternatives to the soporific collegiate lecture, and it is these additions that have made Number Theory such an enduring success.

Through the early 1990s, Pomm and Timmer taught Number Theory together for one session each summer. During this time, they honed their craft, refining skits and adding in-jokes. At the same time, Number Theory's reputation grew, and soon, demand for the course exceeded its limited supply. During the middle of the decade, CTY decided to meet the growing demand by offering Number Theory for both sessions of the summer. Unfortunately, Timmer could only manage to come to CTY for a single session, so extra help was needed. In 96.2, Number Theory was taught by Steinfeld as sole instructor with a TA, incorporating many of Pomm and Timmer's in-jokes and skits, marking the first time someone besides Number Theory's creators worked with the course. In 97.2, Pomm returned aided by a young teaching assistant named Teper. Teper soon became a fully-fledged co-instructor, and as the demands of Real Life drew Pomm and Timmer away more and more, other new co-instructors soon followed. To date, Number Theory has had nine co-instructors, who have worked together in various pairings. Not content to merely carry the torch, these newer instructors have made their marks on the course as well, adding a new skit and compiling the scripts, which in the past were sometimes quite disorganized, possibly even improvised. But Number Theory has not been completely given over to the new instructors: Timmer taught for one session every summer since the course's inception untill 2007, and in 2002, Pomm returned from a three- or four-year hiatus to join his original partner in mathematical crime. In 2008, one session of Number Theory was taught by an instructor who did not use Pomm and Timmer's skits. In 2009, the course was expanded to have two sections per session; one section, that taught by Pomm and/or Brahm and Dimby, uses the "classic" THEO curriculum while the other does not. In 2010, 2011, and 2012 Session 1, the B section (mainly nomores) was taught by Pomm and Dimby with the "classic" THEO curriculum and the A section was not. In Session 2, there was only one session, which despite having Dimby as TA, was not taught with the "classic" curriculum. Dimby says Pomm only teaches Session 1, so it may be advisable to take THEO Session 1 as a nomore, if possible (in 2012, there were so many nomores taking THEO that some of the younger nomores ended up in THEO A, so there is no absolute guarantee of getting into THEO B). Pomm did not return for 2014. In 2016, Pomm and Dimby returned for second session instead of first.

## The Skits

Although the curriculum is important in Number Theory, it is the fun-filled presentation of the material that makes the course memorable. At the core of this presentation lie the Seven Skits of Number Theory. These skits are typically preludes to more serious lectures, and serve to introduce topics ranging from "What is a Formal Proof?" to...well, herein lies a problem. The element of surprise is an important part of many of the skits, and so the instructors are often quite secretive about them. (Interestingly, two of the Number Theory instructors have also worked for the NSA...coincidence?) They tend to speak in code when discussing skits in places where students might overhear, so I dare not say much about the names of the skits, or their topics.

The skits are consistently funny, and surprisingly high in actual mathematical content. Pomm, Timmer, and the other instructors have shown considerable skill in crafting elegant and unexpected metaphors to convey difficult mathematical topics, in such a way that the students might not even realize that they're learning something. The skits also draw on a wide range of props, from brightly colored chalk to fake beards to a tuxedo. Many of the skits are centered on important mathematicians, such as Euclid or Gauss, and a familiar refrain from these skits is, "What's not so well known about [Name] is that he was also..."

Happily, levity in the classroom is not restricted to the skits. Many class jokes vary from year to year, but others have persisted. Several props retain humorous significance outside their skits. The instructors play tricks on their students, sometimes with disastrous results. On one occasion in 2002, Pomm and Timmer, hoping to show their class that a familiar (true) fact was not as obvious as it might seem, unwittingly (and nearly irreparably) convinced them that it was utterly false, and they required the better part of an afternoon to disabuse them of this new-found notion. But perhaps the most frequent running joke in Number Theory is the naming of the theorems. Nearly every result proven in class is given a name that is not found in any textbook. Some are named after students who suggested the particular result, (and so every student can go home and say that she got a theorem named after herself) and many, many other names have a gastronomical theme. This unusual nomenclature can be an obstacle when discussing number theory with someone who followed a more traditional course of study -- you don't know what the Chinese Remainder Theorem is, and he's never heard of Jennifer's Magic Lemma -- but it is always a source of amusement in the classroom.

## The Students

Because Number Theory has both Geometry and Algebra II as prerequisites, the course mainly attracts older students. A large number of the students are nomores. In general, these students have spent a number of years at CTY, and feel quite at home in Lancaster. Also, because Number Theory is such an advanced subject, lying well outside the high school curriculum, students who sign up for Number Theory are truly interested in learning math; nobody gets stuck with THEO as their third choice. Perhaps their prior CTY acculturation, combined with *a priori* enthusiasm, leads them to be more receptive of the instructors' wackiness than a group of randomly chosen squirrels taking, say, Math Sequence.

## The Instructors

Number Theory is one of the more difficult courses to teach: it requires extensive preparation, thorough mathematical knowledge, strong acting skills, occasional improvisation, and a sense of humor. Thus, every math instructor and his pet monkey seems to want to teach it. Unfortunately, teaching Number Theory is sort of like the Mob: in order for you to get in, someone else needs to die. So far, there have been no actual fatalities among THEO instructors, but graduate school, unaccommodating jobs, and other vagaries of life have provided temporary deaths that allowed thirteen instructors and one TA to join the Number Theory club. The following twelve instructors have taught the "classic" version of Number Theory utilizing skits:

- Pomm
- Timmer
- Steinfeld
- Teper
- Quimby
- Sideshow
- Stepper
- Zanmi
- Tortoise
- Birdie
- Brahm
- Dimby

These instructors have been paired up (96.2, 97.2, 07.1, 08.1, 08.2, 09.1 and 09.2 were the only sessions with teaching assistants rather than two co-instructors; Dimby assisted Pomm and Brahm in 2008, 2009, 2010, 2011 and was promoted to instructor for 2012) in many different ways over the years. You might also notice that every instructor is known by a single nickname. This is partially by design: Pomm and Timmer were so called before they started teaching Number Theory, but "Pomm and Timmer" sounded so good (if a bit derivative of a famous magic act) that "Teper" was a natural replacement for "Timmer," and when other instructors joined the fold and started pairing up in multiple arrangements, the "P&T" pattern was abandoned, but nicknames remained a requirement. Though many other staff members are known almost exclusively by nicknames at CTY, the THEO instructors have the possibly unique distinction of having their nicknames listed in documents as official as Lancaster site directories.

Most of the above instructors are long-standing veterans at Lancaster. Pomm is quite simply the most experienced staff member still working at Lancaster, (Pomm and Timmer's CTY experiences both go back to being students in the mid-1980s. At last count, Pomm has been at Lancaster for 35 sessions) and adding up the combined number of sessions all eleven have worked (including other math courses, such as Math Sequence, Math Modeling, and Cryptology) would be a depressing exercise for anyone else who fancies himself a seasoned veteran. For the last few years, Pomm has only taught Session 1. Pomm and Timmer recently released a book on number theory loosely based on some of the CTY Course's skits.

There have been two other Number Theory instructors in 2008, 2009, 2010, 2011, and 2012, but they stuck to a more traditional lecture-style math class instead of relying heavily on skits. Most classes are now taught in this style, as Pomm has not returned to session one in recent years. Instructor Ali Mohajer (aka Shiva, Destroyer of Memes) uses Pomm and Timmer's book for his class, though, so their legacy lives on.

## Concussion

Number Theory is a wonderful course. In over a decade, Pomm and Timmer's skit-based class has grown to be the crown jewel of Lancaster's mathematical offerings, and rightly so. Its unique blend of mathematical instruction and witty humor has left its impression on hundreds of students, and a number of staffers as well. It often boasts some of CTY's most experienced instructors, and a brilliantly presented curriculum. If math interests you, then Number Theory is the course to take -- if you can get in. The author must confess a personal bias, however: it was Pomm and Timmer's magic from his student days that inspired him to return to teach at CTY.

## Past Classes

**THEO.A.LAN.16.1**: This class was compromised of a lot of younger students (who were all actually interested in number theory) and seven nomores and nevermores who had signed up with the expectation of having Pomm and Demby. Instead, they were stuck with Instructor Rob "Ribbity Robbity Broccoli Rabe Rube Rubber Soul Meme" Sulman and TWO TAs- Aaron Dardik (who was mysteriously fired halfway through session, resulting in many conspiracy theories) and later Abby Richards. Rob often yelled at students for unjust reasons, and even caused panic attacks on more than one occasion. He was terrible at explaining concepts and answering questions. He forbade the use of calculators in the classroom. Some number theory was learned, but the majority of class time was spent:

- making up nicknames for Rob (see above)
- making bets with Rob and wagering minutes of break time (Rob almost always lost these challenges, resulting in a ridiculous amount of break time)
- singing a song in loving memory of Aaron ("Aaron Did Nothing Wrong", to the tune of "The Star-Spangled Banner")
- sleeping, often curled up under a blanket with another member of the Back Row Number Theorists (these were the nomores who Rob most often singled out for "extra help")
- reenacting of memes such as "The Scalene Triangle"

Rob made the class count off using Mersenne Primes instead of regular numbers. It was said that Rob had completed the 55-Year Challenge (see "Three-Week Challenge" or "CTY Challege"). He often admonished the students for not getting enough protein in breakfast, to which the students responded by bringing a few dozen hard-boiled eggs from the dining hall and decorating them for CTY Easter. On the glorious evenings when Rob did not attend study hall, the students often raved on the classroom table or slept in corners, until Abby mentioned "pineapples" (the code word that signaled Rob's arrival), at which point all cell phones and calculators would disappear and notebooks would open to a random page in a half-assed attempt to look productive. At the end of session, Oliver and Victoria both took about forty minutes writing their instructor evaluations, with several paragraphs each dedicated to praising Abby and damning Rob.

Back Row Number Theorists: Ariel Uy, Sophia Hager, Drew Genuit, Victoria Provost, Oliver Pennington, Justin Louie, and Nick Vucelic

*If p, a prime, does not divide a*

*Which is a positive integer,*

*Then p divides a to the p*

*Minus one minus one!*

*Is the converse also true?*

*No, not necessarily!*

*Does that thought make you feel blue?*

*No, 'cause pseudoprimes are cool!*

**THEO.B.LAN.17.1**: For this session, THEO-B (aka Beo Thee) was made up of 15 nomores, 1 onemore, and 17 meme gods (including instructor Ali Mohajer). Class consisted of collective suffering, TA Michael's tendency to go clubbing with the amish, shiva (destroyer of memes), double dabs, beach balls, kahoot, Vaughn's 16.1 roommate, and a whole lot of lemmas. Students bonded over the difficulty of the class, but luckily math is all about group work, so we limped through together (though some carrying more weight than others). Overall, Beo Thee 17.1 was a positive last experience for most if not all nomores that were a part of it.