Graduate Course in Number Theory

I just completed my second graduate math course. The subject was number theory which is known as the "queen of mathematics." Before I describe the course I want to briefly mention that last summer I toured the old growth redwood forests in California. When I walked through the redwoods I felt like I was in an enchanted forest. I marveled at the towering sequoias which ranged in age from 1,500 to 300 years old. Each tree is a monument to nature herself and the forest has an eternal quality to it.

I found my number theory course was similar to visiting the ancient forest. We learned theorems from Euclid (e.g. the infinitude of prime numbers) and Diophantus that date back over 2,000 years. We learned venerable theorems from Fermat, Euler, and Gauss that date back over 300 years. Each theorem stands as a pillar of mathematical truth for all eternity.

I have to admit, though, that I was lost in the forest of number theory at times. At certain points I could understand individual theorems and their proofs, but for long stretches I stumbled along. The journey began with chapters on divisibility and congruences. These are straightforward topics that can be taught to bright high school students or the brightest middle school students. Fermat’s Little Theorem and Euler’s generalization were covered. Of course, the totient function f(n) was defined (the count of numbers relatively prime to n) and in a later chapter the remarkable formula was proven. Translated into English this states that the sum of the totient function ranging over all of the divisors of n is equal to n itself (futher translation not available).

Okay they were the easy chapters. Now we move onto quadratic reciprocity and quadratic forms. Our textbook An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery mentions that "Gauss discovered the quadratic reciprocity law just before his 18^th birthday. After a year of strenuous effort he found the first proof, in 1795, at the age of nineteen." I find this fact comforting. The world’s greatest mathematician had to struggle for a year to find a proof for a fact that he knew to be true. So it’s okay to struggle with math homework – that’s its purpose in life.

Binary quadratic forms have the form f(x,y) = ax²+bxy + cy². Mathematicians have studied BQFs intensely and have a complete understanding of them. I can’t say that I have a complete understanding but I did learn a technique based on BQFs that can be used for the following arithmetic parlor trick. Given the prime number 398417 find two numbers whose squares sum to 398417 (answer: 631² + 16²).

For some reason our teacher decided to skip Chapter 4 which has marvelous results like the Moebius inversion formula (don’t ask). So we continue with the chapter on Diophantine equations, the most famous of which is Fermat’s Last Theorem (there are no integer solutions for xⁿ+ yⁿ = zⁿ where n > 2). This leads to an area of mathematics I had never learned before called elliptic curves (these are not elliptical curves). The geometric analysis of these curves (chord and tangent method) yields additional solutions to the Diophantine equation on which they are based. In our class we visit the foothills of the mountains that Andrew Wiles conquered in his 1993 proof of Fermat’s Last Theorem.

We skip chapter 6 to get to the chapter on Continued Fractions. I’m in the embarrassing situation of having to tell people that I’m studying fractions in graduate mathematics, but it’s worth it. These fractions are small wonders and the basis for new insights into irrational numbers. I also learned of Pell’s equation and was able to fully understand an anecdote relating to the Indian mathematician Ramanujan. One of his friend’s gave him a puzzle to determine a house number that met certain conditions. Ramanujan was cooking at the time, but he was able to instantly state the result in general terms using a continued fraction based on Pell’s equation.

For the last two classes our teacher covered eclectic topics in multiplicative number theory. At the very least I learned what Dirichlet’s series are and the Reimann zeta function in particular. Oh to solve the Riemann hypothesis relating to the roots of the zeta function!

Note on textbook: It seems that the textbook by Niven, Zuckerman andMontgomery is widely used as a graduate text. It has survived the test of time having endured five editions since 1961. My belief is that it was a fit text in its early editions, but by the fifth edition it has grown flaccid and overweight. The authors give the most succinct proofs and even skip steps at times.

If I felt stultified by the topics at hand, the authors helped me achieve that state.

The Beebee Gun -Or- What Was I Thinking?

"Dad, can I have a beebee gun?"

"No, you can't."

"Why not? Mike has a beebee gun."

"I don't care. Beebee guns are dangerous and you can't have one."

This conversation was repeated many times when my son Tony was in fifth grade. I'm usually one who can put his foot down and say "no" but Tony found a soft spot in my defenses. He promised to raise his grades if I got him a beebee gun.

"Well you just missed honor roll the last marking period. If you promise to try harder I'll get you a beebee gun."

"I promise."

So I purchased a beebee handgun and let Tony use it in our newly finished basement. Even though I insisted that he use some sort of backstop I started to find beebees embedded in the drywall. I thought "what he really needs is a huge backstop." So I went and bought three sheets of plywood and attached them together via hinges so that they can be bent at an angle to form a backstop, much like a dressing curtain.

The shooting continues and I start to find embedded beebees nowhere close to the backstop. I warn Tony that unless his aim improves the beebee gun is gone. He makes his usual promise and I let the matter drop.

The next thing I know he has several of his friends over and passes around the beebee gun. They must have had a first class shoot-out because I find beebees in the ceiling tiles, the drywall, in the flourescent light enclosures. I also find that the TV took a glancing hit that took a 1/4 inch chip out of the screen.

Okay, that's it, I confiscate the beebee gun (actual picture shown above). Tony and I now spend some quality time together digging out beebees, spackling the drywall, and collecting hundreds of beebees from the all nooks and crannies of the basement.

Now dear reader you must be saying that I'm a complete fool, and I can't deny it. I wanted some sway over Tony and I thought the beebee gun just might do the trick. Of course Tony did not make the honor roll the next marking period.

Even today, many years later, I come upon a stray beebee that escaped my determined effort to remove all evidence of my folly. And when I do, I hold it up to the light and say to myself, "What was I thinking?"

First Graduate Math Course

This is my first post as a graduate math student having completed my first course at Villanova University. Since I was eager to start I took an eight week summer course where each session was 3.5 hours long.

It was intense. The first homework assignment took between 15 and 20 hours to complete even though I skipped problems that were too hard or too time-consuming. Fortunately, the homework subsided in later sessions and even the sessions near the end of the term let out early.

We used what must be "the" book on the subject - Cryptography Theory and Practice by Douglas Stinson. The breadth of mathematics covered in the book (and we covered only about 20% of the book) is amazing. Linear algebra, abstract algebra, probability, and above all number theory. Euler considered number theory to be the queen of mathematics. It is certainly the linchpin of modern cryptography.

My favorite parts to this course were the programming assignments and the study of the RSA public key cryptosystem. I wrote the programs in Java 6 and learned new language features in the process. I'd estimate that I wrote about 1,000 lines of code. My teacher commented that I wrote "nice programs." Damn straight. I've been programming for over 35 years.

RSA public key stuff is fascinating because it's pure number theory. To study it you dip into many number theory topics including primality (this means "is it a prime number") testing and factorization of huge numbers. Now wonder Rivest, Shamir, and Adleman won the Turing Award in 2002. What I don't understand is why the El Gamal cryptosystem became the basis for the Digital Signature Algorithm standard instead of RSA. RSA is just as good for this and it means you have to purchase only one package for your business.

Another thing I don't understand is the theory behind the Index Calculus algorithm for computing discrete logarithms, namely the change in the modulus from p to p -1 as you go from (1) to (2) below.

(1) a^x_j º p₁^e_1j p₂^e_2j ... p_B^e_Bj (mod p)

(2) x_j º e_1j log_a(p₁) + ... + e_Bj log_a(p_B) (mod p-1)

Kind reader. If you can explain, please post a comment with the explanation. My teacher claimed that it was a direct result from Fermat's Little theorem, but I still don't see it.

Dick Cheney Goes to Heaven

It’s just like I heard it would be like. I see myself lying prostrate on the floor. I see people giving me CPR, others calling for help. I hear someone say “it’s too late, he’s gone.” “That’s okay,” I think. “I’m glad the chest pain is over.” Then I thought to myself, “They better give me a good, Reagan-level funeral” as the scene dissolves.

So here I am standing by a desk with a man who looks too busy to notice me. He looks like the late Senator Everett Dirksen, very distinguished, very authoritative. I admire his dark blue suit, his red tie, and especially his American flag lapel pin. I see a nameplate on the desk – Senator Peters.

“Hello,” I say in my deepest, most gravitas laden voice, hoping to command some respect from this lackey.

He puts down a paper, looks up, and smiles. “Welcome, Dick. We’ve been expecting you for some time now. Your resume is remarkable. You should have no problem getting in.”

“Getting in?” I ask. “Is there anything I have to do? Is there anything that I haven’t already done?”

“Not really, Dick, just a preliminary interview. That will be it.”

Senator Peters: “Your service to the American people is wonderful. You were Chief of Staff to President Ford. You represented your state in Congress for 10 years. You served as Secretary of Defense. You were Vice-President during one of the most trying periods of time in United States history.

DC: “Thank you sir. I’ve very proud of my service to America. If you didn’t know, I received the Presidential Medal for Freedom in 1991.”

Senator Peters: “But you never served in the Armed Forces, especially during the Vietnam War.”

DC: “Well, like many young men at the time, I was in college and sought a student deferral. After college, I received a family hardship deferral.”

Senator Peters: “If you never knew the terribleness of war firsthand, how do you feel about sending America’s best into harm’s way? Seems that you’ve done that twice in the Persian Gulf.”

DC: “I know it’s a tremendous responsibility, but I know what’s best for America.”

Senator Peters: It seems that you made up your mind about Iraq long before there was enough evidence to wage war against it.”

DC: “Saddam Hussein was a brutal dictator. Our intelligence reports stated that he had links to Al-Qaeda and was cultivating weapons of mass destruction. Anyone who disputes this just doesn’t know the facts.”

Senator Peters: “It even appears that you benefited from the war with your ties to the oil industry and the defense industry.”

DC: “Nothing could be farther from the truth. Those accusations typically come from my political opponents. I have not benefited for any activity that I believe is in the best interest of the United States of America.”

Senator Peters: “Well, as I said at the start Mr. Vice President, your resume is impeccable. Congratulations and welcome to heaven. Yours is the first door down the hall on the right. It’s the red door.”

I see the red door down the hall and walk toward it. It’s this easy then, entering heaven. Christ, I deserve it. With all of the great service I gave to the good old U, S, of A. Working for that idiot Ford, never reaching the top, and then working for idiot #2, Bush, for eight years. Thank god that they listened to me, and did what I told them, those fucking idiots.

I open the door and walk in. I notice that I’m wearing a warm-up suit and that I feel fine. No chest pain at all. I also notice that there’s a sweet smell of incense in the room and there’s strange, foreign music playing. Not exactly what I thought heaven would be like, but not bad either.

The room’s kind of dark and it takes my eyes a little time to adjust. To my surprise I can see a giant round bed fitted with a scarlet bedspread and silken sheets. Then I notice the mirror over the bed. This looks like a scene from playboy mansion. What next? Girls?

Oh my god, it is girls. Young, voluptuous girls enter from a side door. So many I can’t count. Each is dressed in silk gauze. Each is dark-eyed, with a dark complexion, though I wouldn’t call any of them “black.”

They file past me and every one either smiles at me, or purses her lips, or lifts her bosom with one of her hands. Christ, am I going to get laid here? Looks like this Wyoming cowboy is going to shoot off his pistol a few more times.

So I look over these girls. None of them says a word, but their body language speaks volumes. Half of them just sit on the floor with their legs open wide and grin at me. Some look young enough to be my grandchild. Others are older, but not nearly as old as my wife. Choices, choices.

Okay, I’ll take you as I point to teenage beauty. She squeals in delight, and I heard a sigh of disappointment from all of the others. This beautiful creature literally skips around me as we head to the round bed. By the time we reach the bed, she’s naked. All I have to do is drop pants on my jumpsuit. I’m amazed that at my age, I’m hard as a rock and ready to go.

So we’re both down on sheets, which are kind of slippery. I finger her and see that she’s wet as a mop. Since she’s ready, I’m ready. I’m going in and … oh my god that hurts. It feels like my dick was just cut off.

Next thing I know I’m sitting at the desk of Senator Peters again. He’s shaking his head.

Senator Peters: “Dick, you failed the interview.”

DC: “What? We had the interview. I did great.”

Senator Peters: “That was the preliminary interview. We knew that you were good with words, a veritable prevarication machine you might say. So we put you to the test, the silent interview. You failed just like the rest of them.”

“We can now call you on all of your bullshit. You were so good on earth that we had trouble telling the bullshit from the non-bullshit. You were so convincing that even from our vantage point, we couldn’t tell if you were sincere or not. Here, I'll just tell you what we've got on you.”

“You help found the ‘Project for the New American Century’ which advocated the U.S. use its military strength as an instrument of foreign policy.”

“You directed the Energy Task Force and published a report that stated it is in the US interest to manipulate oil rich countries to secure future oil supplies.”

“You were the Bush spokesman for the invasion of Iraq and the removal of Saddam Hussein stating repeatedly that:
- he was developing weapons of mass destruction
- he was an ally of Al-Qaeda
- he supported the September 11 attacks against the US.”

“You saw to it that Halliburton Corp. received $7B no-bid contracts following the US invasion of Iraq.”

“You, sir, are a warmonger and a war profiteer. Goodbye Mr. Cheney,” says Senator Peters. He pushes a button under his desk, and just like in the movies, a trapdoor opens under my seat, and I fall into darkness. And as I fall I feel it getting hotter and hotter. “Christ, what’s happening,” I think. And it keeps getting hotter and hotter.

The next thing I know I’m lying of the floor of a cell with harsh lights shining down on me. There’s a mustached man with army fatigues standing over me. The bastard kicks me with all of his might in the gut.

“Hello, Cheney. Welcome to hell,” Saddam says.

The Recess Posse

This is a true story from my youth. I haven’t shared it before now.

In eighth grade all of the "good" boys, as deemed by the school nuns, were impressed into the school's safety patrol. So I was picked to be a safety.

We were required to conduct ourselves in a morally fit manner, and were given specific assignments. My assignment was particularly bad -- I was to oversee the first grade class during lunch recess in the rear of the school.

I hated this. I was separated from anyone close to my age, and I had to enforce the nun's rules concerning recess. First and foremost, the first graders were not allowed to run! Heaven forbid they might trip and fall on the asphalt. So for the first two weeks I was constantly chasing after the violators and making them do a time-out by standing against a chain link fence.

It didn't take long before I noticed that it was the same bunch of kids that I was punishing day after day. Since I was going crazy enforcing the silly rule, I decided to make my own rules. I rounded up all of the "bad" first graders and deputized them. Of course, they wanted to know what that meant, so I told them that they were on my side and when I gave the order, they had to run after any kid they saw running and bring him to me.

What a great idea this turned out to be! Not only did I give them a chance to exercise, but I had the most enthusiastic enforcement team you could imagine. If someone outside the posse started to run, the posse would collar him right away. It was so effective that after a while, to relieve boredom, members of the posse would just start running to see how long they could elude their friends.

It never occurred to me that if the nuns found out what I was doing, I would have been severely punished, maybe even suspended. As luck would have it, I received a promotion and changed assignments. This wasn't due to the taming of the running lust of first graders, but rather to the fact that the captain safety was kicked off the force. He was caught talking during morning prayers and was instantly dismissed by the eighth grade nun.

Everyone near the top of the ranks was bumped up one position. I became lieutenant safety and was assigned the bicycle rack at an entirely different area of the school.

My story was never told till now.

Bad Line Call

Tuesday night is tennis night, and this one was just like any other tennis night, except that it was different. It was the last tennis night of the contract season, and I was playing my tennis nemesis, B. I expected to lose to B. that night; I just didn’t know how badly I would lose. I expected to lose because I always lose to him. It’s not that B. is in a totally different league than me, like he’s a tennis pro and I’m a rank amateur. We’re in the same league, and that’s what so frustrating about the fact that he always wins.

B. is a player who instinctively knows how to play the game. He knows how to hit all of the strokes. He knows when to come to the net, and when not to come to the net. He knows how to position himself when the ball travels to the opponent’s side of the court. His tennis instincts are as strong as a predatory lion’s instincts are when it comes to stalking an antelope.

If B. has a weakness it’s his lack of conditioning. Since our contract time is limited to one hour this isn’t a big factor. In fact B. compensates for it by his sheer will to win. Let me rephrase that – B. refuses to lose. He pushes himself to the limit whenever defeat is near. It doesn’t matter whether it’s losing a set, a game, or a single point. B. refuses to lose.

But something was different that night. We were tied 4-4 and I was holding my own. I had gotten to this score on occasion in the past, when B. would suddenly realize that he might lose. B. would focus with incredible intensity and his serves would blaze into the corners of the service box. If his service didn’t win the point, his groundstrokes would, always attacking my weak backhand until I either hit a short ball or a shallow lob, either of which was enough for him to put away the point. Of course, I knew all this and it would give me a case of the nerves just like an antelope gets when he sees the lion.

B. served in the ninth game. I could tell he was tired because he started to serve and volley. He wanted the points to be short like the quick kill by the lion going for the jugular. I just kept lobbing my returns and instead of sailing long, they were landing deep. This not only frustrated B., it also exhausted him as he sprinted back to make a return. With the game tied at 30 – 30 we start trading groundstrokes, moving each other from side to side, each hoping to hit a winner and end the point. B. hit a deep drive to my baseline. I couldn’t catch up to it, but I could watch it closely. I saw it land squarely on the line, and I immediately call it “out.” B. was too far away to challenge; he just bent over to catch his breath. On the next point he opened his forehand and the ball sailed out of bounds. I had just broken his serve.

I served out the last game. On set point B. was totally spent. After he returned service he came to the net to rattle me as he had done so many times before. I lobbed him and he didn’t even try to make a save. He just turned and watched the ball land in bounds. I had won my first set in 10 years.

But what about that bad line call? I knew it was the wrong call as soon as I made it. Since then I have thought long and hard about it. If he had won that point, he may have won the game and eventually the set. The lion feasts again. I thought of all of the injustices I had endured. Doesn’t it matter that B. foot faults frequently when he serves? Doesn’t it matter that B. enjoys trouncing me 6-0, 6-0 when I’m playing poorly? What about the times he gets his hot serve going and deliberately picks on my pathetic backhand return? And he keeps doing this through both sets of the contract hour. I know these are all rationalizations, serving me, the antelope. What would your call be?

The Handshake

The plane had landed at Philadelphia International Airport and had rolled to a stop outside of the gate. I knew that this was the time to make my move. I stood up and leaned over the mammoth figure still reclining in his seat. He looked at me crossly. “Sir,” I said, “you are the greatest basketball player of all time.” I stuck my hand out forcing him to reciprocate. “Thanks man,” he mumbled as he shook my Lilliputian hand.

Now how can you sleep when the greatest basketball player of all time is sitting directly in front of you? I couldn’t. All I could think of was what he could still do on the court against the Sixers new big man, Darryl Dawkins. The Dipper would just eat him up, I thought, on offense and defense. But that’s a fantasy. Wilt will never set foot on a court again. Maybe I’m dreaming this. After all it’s the red eye from LA to Philly and you’re supposed to sleep.

What can you say about Wilt Chamberlain? Most sports fans know that he scored 100 points in a single professional game. But how about the 50+ points per game he averaged for an entire season? Or the record 55 rebounds he grabbed in a single game against Bill Russell. Or the only time a professional player scored a double-triple-double – 22 points, 25 rebounds, and 21 assists. These records are as impossible as an asthmatic climbing Everest without oxygen.

In his early days he was unstoppable. The NBA felt it had to at least slow Wilt down so they changed the rules. The three-second violation, widening the foul lane, and both offensive and defensive goaltending rules were instituted to throttle The Dipper. They didn’t.

Why am I thinking about this? I’m supposed to get some sleep. Don’t you think Wilt’s sleeping in his seat?

As I board the plane late in the evening, I’m surprised at the check-in counter. I have just received a free upgrade to first class. I enter the first class section and see that there are more seats than people. Now I can really spread out, relax, and sleep. But wait a minute, who is this towering figure now boarding at the last minute that everyone seems to know. My God it’s Wilt Chamberlain, and he’s sitting in the row in front of me! How like Wilt to hold up an entire flight until he’s good and ready to fly. After all he’s probably the only profession coach who failed to show for his team’s games. Now what should I say to him?