September 16, 2006--Saturday Story: "A Matter of Factoring"--Concluded

In Part Two, Lloyd revealed that he had so much difficulty understanding the most fundamental and elementary aspects of Algebra that he attempted to convince his parents to allow him to drop out of school and get a jumpstart on a career as a carpenter. His father countered by suggesting that he consider becoming a pre-med once in college and use his woodworking skills as a plastic surgeon. Lloyd’s mother quickly intervened and arranged for Cousin Chuck to tutor Lloyd. He did such a good job that Lloyd was able to unlock the mysteries of “x” and "the power of equations," and Chuck, also known as “Charlsy,” was rewarded with slice after slice of Icebox Cake.

In the Third and Final Part, we find Lloyd back in Algebra class at Brooklyn Tech, no longer plagued by migraines and . . . .

Back in class at Brooklyn Tech, with my headaches cured and my ‘x-problem’ overcome, I became a demon at solving all kinds of algebraic problems—those involving negative numbers, square roots, adding and subtracting polynomials, and of course factoring.

When Dr. Kaufman asked Aaron Bernard, Joey Lombardy, and me to come up to the blackboard to factor a series of quadratic equations, I did not fear public humiliation. In fact when he wrote--

x2 + 4ax + 3a2

--almost as fast as I could move the chalk across the board I factored it to yield—

(x + 3a)(x + a)

“Well done Dr. Kaufman said even before Aaron much less Joey, unaided by his accomplice Charlie Rosner, were able to write two sets of empty parentheses.

And then when Dr. Kaufman scrawled--

n2 – 12n – 35

--as the second quadratic for us to factor, as Aaron picked perplexedly at one of the pimples on his cheek, I grinned toward Dr. Kaufman, my braces glinting in the fluorescent light, and said, “It can’t be factored because it’s a Prime, and Primes can’t be factored.”

“Excellent,” he said. “Along with Mr. Rosner, please see me after class so we can talk about the Math Team. The JV of course.”

Behind me I could hear Milty Leshowitz begin to hyperventilate. “It’s my asthma,” he gasped and ran out of the classroom toward the nurse’s office, which up to then had been my refuge.

* * *

Charlie and I remained uncomfortably at our desks, averting our eyes, as the rest of the class scattered at the first sound of the end-of-period bell. Though we craved the distinction Dr. Kaufman’s invitation to remain bestowed upon us, we had mixed feelings about being thus set apart from our fellow classmates.

This was to be a perpetual struggle at the highly competitive Brooklyn Tech, where at the end of each term everyone’s cumulative grade point average, down to the third decimal point, was published, in bold type, for all to see and compare in the Tech student newspaper, The Survey. We struggled to both compete with classmates, who literally sat to our left and right, while at the same time attempting to remain friends. Against these “friends” we contested to place higher on the class-standing list where the difference of just one place would determine if you got a free ride upstate at Cornell while they were reciprocally exiled to the concrete campus of City College.

And, of course, the friend on your right was eyeing you in exactly the same way—he saw your crawling up out of the subway in Harlem as assuring that he would frolic on the green hills of Ithaca. It was all an equation, a balance—to rise on one side, something on the other had to descend.

The one thing our English teacher, Miss. Ryan, was able to get us to pay attention to was her warning not to help anyone with homework. If we did so, and as a result a classmate got one point more than you on the grammar exam, that measly single point might mean that he, and not you, would wind up in the top ten or highest hundred on the GPA list.

“And you know the implications of that,” she would intone portentously. If we did help a fellow student, she warned, and he as a result surpassed us as “the beneficiary of that selfless generosity,” there could still be lessons to learn since that benevolence and its “ironic” consequences” could serve as “a metaphor for life itself.” Though as an English teacher she was always talking about things like “irony” and “metaphors,” since these were concepts that very few of us understood, we didn’t think too much about the “irony of benevolence” but rather continued to focus on something we could grasp--how to avoid splitting infinitives, an unforgivable transgression to her. Life metaphors could wait until our junior year.

When all our Algebra classmates had departed, Dr. Kaufman summoned Charlie to the front of the room. I did everything I could to force myself not to listen in on what they were saying, feeling whatever it might be was private and should thus remain between them. However, in spite of these noble intentions, I still found myself inexorably straining forward in my seat so I could take in every single word.

Even if less-than-conscious, I felt impelled to do so, as if by a force outside myself, thinking that whatever might transpire between Charlie and Dr. Kaufman would be similar to what would be expected of me; and, in my competitive mode, after all wasn’t I being trained for that at Tech, I wanted to take advantage of anything that came my way that would give me even the slightest edge.

It was, though, not so easy to hear them since Charlie, tall enough for the basketball team as well as smart enough for the Math Team, Charlie towered over and shrouded the seated Dr. Kaufman, who, behind his battered desk, thus seemed so reduced in size and shrunken in stature. But it turned out not to matter since Dr. Kaufman did not ask any questions or quiz Charlie in any way; he simply told him to show up for the tryouts next Wednesday, at 4:30, in the Math Team office on the first floor, right next to the nurse’s office. About this, I would not need directions.

And so, when he signaled to me, I approached him without trepidation, assuming that he would simply tell me as well about the time and place for the tryouts. But Dr. Kaufman nodded at the old library chair beside his desk, and I sat down. My heart began to thump. I felt it throbbing all the way up in my throat. He looked at me without saying anything for what felt like fifteen minutes. I was beginning to experience palpitations.

“Lloyd,” he said, using my first name—prior to that he had only once before called any of us by anything but our last names, except when exasperated with Milty, when he called him “Morty”—“Do you remember the other day when I quoted Don Zagier’s thoughts about Prime numbers? From, I think it was, from his inaugural address at Bonn University, when he became one of the directors of the Max Planck Institute?”

“I remember, what you said Bernhard Zagier wrote about them. About their distribution.” I couldn’t believe, without restraining myself, I had appeared to be correcting Dr. Kaufman.

“Ach, good,” he laughed, “I forgot I called him that. We in the field know him more familiarly as ‘Don.’ He’s an American you know.” I nodded as if I did. “Well, Don had more interesting things to say about Primes than almost anyone. And I am telling you this because you appear to be interested in them and are even beginning to show some signs of promise. Talent I do not as yet know about.” I began to fidget. Which he noticed. “That is all right. We will know soon enough. That is why you are here—for us to find out.” This sounded both encouraging and ominous.

Without waiting for me to say anything, he continued, “Don, I felt, was always too optimistic, perhaps even a little arrogant when he claimed that ‘though they grow like weeds they exhibit surprising regularity.’ Do you remember that?” I did. “Good. Most mathematicians through the centuries struggled to find that regularity, what Don called their ‘precision.’ From the earliest days in Greece and Arabia.” Inexplicably, images of Cousin Chuck gobbling Icebox cake flashed through my mind. “Struggled unsuccessfully, I should add. Don as well. Indeed, I understand that he is still trying and continues to write elegantly on the subject.” Dr. Kaufman had again returned to the subject of mathematical elegance.

“For example, though numbers are the simplest of mathematical elements they, perhaps for that reason, do you understand, have inspired the lushest prose. You find mathematicians referring to them as ‘mysterious,’ ‘stunning,’ ‘diabolical,’ ‘harmonious,’ ‘the Holy Grail,’ ‘divine,’ even, yes, ‘glamorous.’” His eyes sparkled as he pulled these words as if from the air surrounding us in that barren room.

“This may still be unfamiliar to you, but mathematicians, when considering Primes, they often employ the vocabulary of first love. To them, they are objects of great beauty.” He paused to look at me in a way that made me quiver with excitement and nervousness.

“One day, we can hope, soon, we expect, you will participate in the exploration of these mysteries and taste this chaste form of love.” I was glad to hear him describe it that way, and began to calm down. “You may have already begun to do so, to have touched some of this, instinctually. We will discover, together, if you share some of this gift. When you so quickly, earlier today, perceived that n2 – 12n – 35 is indeed a Prime, a very special Prime, exhibiting its own mystery, its own beauty, I thought, if you are patient and work hard, you will find out.”

I pledged to myself that I would do both.

“Of course, along with Mr. Rosner, you will tryout for the Team next week.” I indicated I wanted to. “But before you leave,” I saw under the desk his hand with the chalk begin to twitch, “I need to tell you something that the great Tenenbaum and France said about our numbers. I cannot recall it exactly, but since it is very important I wrote it down and hopefully have it here in my desk.” With his right hand a claw clutching the chalk, he rummaged around one-handedly searching for it.

“Ah, here it is.” He looked up at me with a broken-toothed smile, and read, ‘As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of them is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant.’”

He peered at me again as if he might find within something worth understanding or knowing. As yet, though I too had recently begun to search, there was nothing worth noting.