# Number Theory

Math Course | |
---|---|

Course Code | THEO |

Year Opened | 1985* |

Sites Offered | LAN |

## Contents

## Course Description

From the CTY Catalog:

Called "The Queen of Mathematics" by the great mathematician Carl Friedrich Gauss, number theory is the study of the natural number system from which all others are derived. Despite the simplicity of the natural numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture, formulated in 1742, which posits that every even integer larger than 2 is the sum of two prime numbers, has defied all proof attempts. In this proof-based course, you’ll learn the major ideas of elementary number theory and the historical framework in which they were developed. While strengthening your ability to analyze and construct formal proofs, you and your classmates will explore topics such as the Euclidean Algorithm and continued fractions, Diophantine equations, Fibonacci numbers and the golden ratio, modular arithmetic, Fermat’s Little Theorem, and RSA public key cryptography. You’ll leave the course with an appreciation for the elegance of theoretical mathematics and the ability to craft rigorous arguments.

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 challenging subject for younger students, even students of CTY's academic caliber. Thus, they sought ways to make the material accessible and entertaining, and 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 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 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 until 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 younger nomores ended up in THEO A, so there is no guarantee of getting into THEO B.) Pomm did not return for 2014. In 2016, Pomm and Dimby returned for second session instead of first. In 17.2, it was taught by Pomm and Dimby. Pomm taught Number Theory for LAN 18.1.

## 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 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 high in 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 students might not realize they're learning. The skits also draw on a wide range of props, from brightly colored chalk to fake beards to a tuxedo. Many 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 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 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 like the Mob: for you to get in, someone else must die. So far, there have been no actual fatalities among THEO instructors, but graduate school, unaccommodating jobs, and other vagaries of life (see Rob Sulman below) 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 the most experienced staff member still working at Lancaster, (Pomm and Timmer's CTY experiences both go back to being students in the early-1980s. At last count, Pomm has been at Lancaster for 44 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 other Number Theory instructors in years since 2008, but some 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.

Rob Sulman was one long-standing Number Theory instructor at Lancaster. As of 2019, JHU has disinvited him from returning to teach at CTY due to many well-documented issues regarding Rob's insensitivity to students with mental health conditions.

## 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 interested in number theory) and seven nomores and nevermores who had signed up with the expectation of having Pomm and Dimby. 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 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 paragraphsdedicated 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.

**THEO.B.LAN.17.2**: This class was made out of 14 boys and 2 girls both named Angela. There were mainly two groups, The #ganggang and everyone else. They loved the skits and the class was never boring. Some inside jokes are Which is Abusrd, Bad banana, Cody's Blammo kill (with rainbow), Max's pitching, "a squared plus b squared equals c squared" (in unison), fake Russian, guess and check, the toe show, "Well that won't work," What do we want? A strategy! When do we want it? Now!, There's something wrong with your dog, Tims frisbee, Proof by math, Xrice, Rock Paper Scissors SAYS Shoot, and satisfying board cleaning. They all loved this class and will miss it forever.

**THEO.A.LAN.18.1**: This class was made up of 11 boys and 3 girls (originally 4, but one switched out to Philosophy of the Mind.) The class was lucky enough to enjoy the teaching style of Pomm and Dimby as they went through three weeks of skits. (The potatoes are excellent today!) With blue becoming orange, Amelia's Euclid, and Kevin's endless stream of worries, this class was not only educational, but highly entertaining.

**THEO.B.LAN.18.1**: 10 boys and 4 girls, mostly nomores, put together the THEO B class this session. The instructor was Rob "Ribbity Robbity Broccoli Rabe Rube Rubber Soul Meme" Sulman, and the TA was Michael Bellissimo from 17.1. The class managed to survive the torrent of lemmas, theorems, definitions and proofs launched at them, although some (like the Chinese Remainder Theorem/5 Hour Theorem, so called because it took five hours longer to prove than it should have) were not quite absorbed, thanks to Rob's infamous teaching skills (see 16.1).

Many of the class breaks were spent on Hartman/Weis field, where the class would play Frisbee, rally with a beach ball, sit down for a game of Coup, or read the almighty masterpiece that is My Immortal. However, breaks tended to belong as Rob would make wagers for break extensions. These varied from listing prime numbers for three minutes straight to a race with Rob himself, which the class won. On one Rob-less evening early in the session, TA Michael sprained his ankle, so the class got an hour-long break.

Because Rob knew he would go on many tangents, he asked the class to fill in a box in the corner of the blackboard every time he got off-topic. One of these discussions was about the mysterious grapefruit that was almost always on his desk, often lathered with unidentifiable cream cheese-like substances and impaled with a KIVO spoon. The class wasn’t sure if the grapefruit was replaced each day or if it was the same for the entire session.

Rob also enjoyed guiding meditation sessions on wholesome topics such as the vegetable family and friendship between a swallow and an elephant; the class convinced Rob to make these a daily occurrence. During these meditations, the class would shake with laughter as they copied Rob’s mystical arm movements (of which many involved flapping like a bird or t-posing, much to the class’s delight and confusion). The class was promised a meditation on the Heimlich Maneuver, but it was never delivered.

Rob seemed to have a strange obsession with the critically endangered big cat species *Acinonyx jubatus venaticus*, or the asiatic cheetah.

All in all, the class learned a little, mostly thanks to TA Michael's endless supply of support and chocolate. They bonded over monkey entertainment, beach balls, and the transitive property (hippity hoppity). They will never forget Fermat’s Little Theorem, due to Rob’s untitled magnum opus to the tune of 'doe a deer' (see 16.1).

**THEO.A.LAN.18.2**: THEO.A was taught by Ali. It was a sleep-filled class, where most kids caught up on lost hours of sleep from previous nights. Not much Number Theory was learned from the instructors teaching style (see: copying notes on blackboard). Nobody cried when the instructor was revealed to be Ali instead of Pomm, but in the end, the students wished that they did. Many students improved their doodling skills greatly, and in the last few days of the class, were able to produce amazing caricatures of fellow students. At least breaks were fun.

**THEO.B.LAN.18.2**: THEO.B (NUMBO B!!!) had the great fortune of being taught by the legend that was Rob "Ribbity Robbity Broccoli Rabe Rube Rubber Soul Meme" Sulman (who celebrated his tenth CTY anniversary) and was TA'd by Dakota White. Rob, a guided meditation aficionado, consummate playwright, and V8 enthusiast, was prone to tangents, as described in previous sections. On the first day, the class spent a good half an hour discussing what was the solution to 0/0. Oftentimes, he would ask Dakota to keep him on track. The class loved him for the wholesomeness that he represented, and his love for the Fermat's Little Theorem song. Breaks were often spent playing Four Square on Hartman, where players could not get out serves, had to play "chill," and could not get Bubbles (who loved the proof for infinite primes) out.

**THEO.A.LAN.19.1**: Taught by dUrIaNeL (Dionel) and TA’d by Chris. This class was made up of Toby Gang and the three girls in Bree’s hall who were honorary members of Toby Gang.

**THEO.B.LAN.19.1**: BEO THEE was taught by Joe (aka Joe Joe, the All Father, or Joehemian rhapsody). This class created an entire subculture of CTY, including positions for each class member, relics, after class chants, and canon songs.

HIGH HOLY CANON

- Man vs. Machine - by Hovan Liu, a member of the class, created using only sounds made by the class. Positions were often based off of the lyrics to this song.

- Bohemian Rhapsody - because Joehemian rhapsody. The official dance includes T-posing and eventually arm circles.

- Mans Not Hot - but specifically the version where Daniel is rapping to it.

LOWER CANON

- Piano Man

- Country Roads

POSITIONS

- Man - Micah

- Machine - Jake

- Mama (UwU) - Olivia

- Gun - Mikaela

- Euler - Ari

- Beelzebub - Jessica

- Papa Joe - Julien

- Whore - Alex

- She - Lindsey

- Fish - Alvin (or Chaim. This is up to debate)

- PHIsh - Chaim (or Alvin. This is up to debate)

- Lamb Sauce - Daniel

- The Dealer/Setmaster - Zelie

RELICS

- The Joe Juice Cup

- The lost (and found) Frisbee

- The whore’s leash

AFTER-CLASS CHANTS

- Bees - What do we want? BEES! When do we want them? NOW! Where do we want them? IN OUR P*****S! Why do we want them? TO FILL AN EMOTIONAL VOID!

- Thursday Night - P***y on my left, salad on my right, ready for a casual Thursday night!

on our last class Joe sat us down and encouraged us to be more appropriate

**THEO.A.LAN.19.2**:
Make CLEAN Groups.

Instructors Pomm and Dimby.

We made the best proofs separately together.

The class was insane and possibly the best thing in the world. The author has taken six CTY courses and this was by far the most fun and insane. Notable moments include the many skits and getting together on the last night ten minutes before lights out to sing/scream songs. Chalk was thrown, and chalk mountain was created. Pomm and Dimby clearly had been doing this for a long time because it was a wonderful experience.

Brick, brick, potato, brick, brick, potato, linear combination, Diophantus! Relatively prime, relatively prime, relatively, relatively prime, relatively prime, relatively, that’ll work, that’ll work, no. That’ll work, that’ll work- yes.

We have a convoy, a great big convoy, trucking through the night, she’s a beautiful sight. We have a convoy, a great big convoy... yay.

The personalities of the students range from AW, also known as the 14-year-old Mark Zuckerberg, to N, who would scream inconveniently loudly every time a proof was finished.

After every "proof" chant, J would hold up a sign and yell "Pika," to the disgust of the rest of the class. It became reminiscent of dances; everyone would yell "NO PIKA" back (or simply boo) in the manner of RA’s replying with "NO ORGIES."

AW and Y kept making...interesting comments. One such example is the proof of q-ness where p=q; this somehow proved that AM has p-ness (which they do not mind).

Pineapples are of great importance. There were approximately seven different pineapples at any point in time in class. On such example of this was R being AM’s pineapple. (it’s a proof. it happened.)

The almighty selfie stick experienced a full character arc through the class, from its origin on the first day to its eventual death after a skit involving a crossover between Diophantus and Fermat.

How are the potatoes doing? They're about one pound.

Orange is blue and blue is orange.

Jeff Hashfield was the best TA ever. I love him with all my heart. Love you babe <3. ------------------- [Disclaimer: This is a joke and this is not encouraging pedophilia in any way, shape, or form. Jeff Hashfield was not in fact our TA, but actually, the greatest mathematician in the world. He even designed Hashematics!]

Every student in this class became fluent in Froggish after continued education from the fathers of the language.

All cows on Hartman Green are purple.

Omega, to the omega, to THE OMEGA, TO THE OMEGA, TO THE OMEGA, **TO THE OMEGA, TO THE OMEGA**

We repeat. Make CLEAN groups.

**THEO.B.LAN.19.2**:
We were collectively known as "Joe's Hoes" in honour of our Instructor Joe. He needed to be taught how to teach after a couple of days, and is a 27-year-old man. He also didn't engage the students. This was an unfortunate situation.

The class consisted of girls from someone's hall (no one cares) and the legendary Cream Taco Men from Saul's hall.

Sickness spread like wildfire for the first two weeks, and although the class never really got better, we collectively got over thinking about it. Also, it became hard to know if you were sick or not because sleep deprivation and chugging whole 1/2-gallon SKL bottles in one day left both Donald and Ryan feeling many feelings, so we were unsure if sickness was one of them.

First Sunday, Adil and Ryan got two strikes and a 30-minute sanction for being in the basement of Steinman.

TA Dakota was cool. She was young, had previously been to CTY, and was fun to have as a TA. She tolerated a variety of strange behavior from us students, so it was nice to have her as a relate-able TA. She also let Ryan go to the nurse to nap.

Other than being a cool person, Sasha had the highest tendency to explode into Mathematical Orgasms (sounding exactly like you'd expect) once a theorem or an idea is understood. She also performed beautiful songs about herself during the talent show.

Other than Donald's quote list, the instructor Joe (the all father) had his own quote list. We saw it on the third Tuesday.

Donald and Ryan bought Joe a 1/2 gallon of SKL and he drank 1/3 of it in class one day. He finished the rest within a week.

Joe was willing to sacrifice teaching the class to chat/joke around with Annalise.

Joe was known for continually yelling "Donald, don't do that," "Donald, why are you doing that," and a variety of other variants.

The majority of slide-show approved images from this class include pictures of Ryan and Donald chugging 1/2 gallon SKL, everyone cult dancing, amongst other things. Slide-show worthy but unapproved images consisted of Donald brushing Ryan's teeth, a duct tape bra Alex made, and some sexy drag day photos of Donald, Kenny, Adil and Ryan.

Joe didn't care enough to encourage us to be more appropriate.

*Donald's Quote List*

- "That's not Math" - Joe
- "The Math builds character" - Joe
- "At least stop squirming like some sort of worm" - Joe
- "You're now sanctioned to the shadow realm" - ?
- "There will be no napping today" - ?
- "What's more important than Number Fields? The answer is nothing." - Joe
- "I shouldn't threaten students. That's probably not good" - Joe
- "InDUCKtion" - Dakota
- "DUCKota" - Dakota
- "Lanyard Bondage!!!" - Alex
- "My gum fell out of my mouth" - Alex
- "I don't know, it (a hoe) kind of looks like a golf stick" - Sasha
- "I'll be wet all day" - Alex
- "I will fully punish myself tonight" - Ty
- "I need more SKL" - Joe
- "Is [a booger] more like Joe or like Donald?" - Pranav
- "Ahhhhhhh" - Sasha, Math Orgasm
- "I'm Gluten" - Sasha
- "I felt your sneeze inside me" - Alex
- "I eat coffee all morning" - Alex
- "That was not a mathematical orgasm" - Sasha
- "My therapist says I'm his favorite because I'm the most problematic" - [Anonymous]
- "Donald, what position are you sitting at?" - Alex
- "I need to hide this sandwich in my room" "Why?" "Because of the goblins" - Alex, ?, Alex