Finding Zero

by Amir D. Aczel

  We walked through the lab. At one table a student painted with modern gold color a wooden Buddha statue that showed traces of once having been gilded. Two students were trying to attach a new pedestal to yet another statue. They pushed and twisted and hammered. We walked over, and Rava gave them some instructions and help. Finally, he announced loudly “It fits!,” and everyone clapped. Where did the pedestal come from? Nobody knew. It had been found in a pile of statue parts that sat outside the lab, Rava informed me. And there was a statue that lacked an arm, and another Cambodian archaeology student was attempting to attach an arm to it—I had to wonder where that arm came from.

  This was not even archaeology as Evans would do it. This was a travesty. I knew from talking with conservators that state of the art methods now emphasize stabilizing an artifact to prevent further deterioration from insects or environmental elements while retaining the integrity of the piece and altering it as little as possible. “When you will come back, you will see how beautifully restored and shining will be the K-127,” Pellegrino said as my tour of the lab ended and she showed me to the door. I thought I discerned a mischievous smile on her face, as if to say, “Something that was briefly yours is now mine!” Or maybe it was my overactive imagination responding to the great excitement and stress of this long day.

  With a heavy heart, I left Siem Reap for Bangkok. In an e-mail message I told Bill Casselman of the University of British Columbia, the mathematician who had first led me to Cœdès, what had happened. He was clearly furious. “What if they take it to Italy?” he fumed. “And from your photos it is clear that the inscription is in perfect condition and has an invaluable patina—if it is destroyed by their work, this priceless find in the history of mathematics will be worthless and lost forever.” He thought it was clear that Lorella Pellegrino now wanted the credit for this find. Casselman offered to write to her, so we both wrote e-mail messages to Pellegrino, in the name of historians of mathematics, pleading with her not to disturb the artifact.

  After a sleepless night for me, on January 3 Pellegrino relented somewhat and wrote that she would not “restore” the artifact. “However, I will study it thoroughly, and report the results to the world of science,” she wrote me in Italian. I took this to mean that she considered the find her own. I was relieved she wouldn’t harm it, as she said. Still, we had to do something to save K-127 more permanently. But what? “Maybe she’ll forget about it, do her other work,” Debra wrote me in an e-mail. I had a feeling that Pellegrino would not stop. She had fallen upon an important find, and I guessed that she would do everything she could to make it hers. Maybe all the restoration teaching stuff was just a smokescreen—an excuse to take possession of the artifact.

  23

  Twelve days passed. And then, it appeared: Lorella Pellegrino had further plans. “New location for K-127” was on the subject line of an e-mail she sent me on January 15, 2013, and the message itself announced that she had moved the giant stone inscription into her own lab and would soon start the work of “studying it structurally and bacteriologically.” She attached a photo of a visitor from Italy, some distinguished professor to whom she showed the inscription, proudly posing next to it. When I sent Casselman the photo, he wrote me wryly, “Does he even know what he is looking at?”

  And I feared what would happen once the “structural and bacteriological studies” began. The stone inscription was perfectly legible and clear. All that needed to be done with it was to place it in a museum so mathematicians, historians of science, and the general public could see it. But Pellegrino refused to give up what she seemed to perceive was a big fish that had just landed serendipitously in her net. She appeared bound to leave her impression, her imprimatur, on this “precious find,” as she called it (using the Italian word prezioso over and over again).

  As if to taunt me (or so I imagined), she sent me messages in Italian every couple of days, describing what she was doing with the inscription. “My students and I have just completed a 3-D study of K-127,” she wrote. And a day later she wrote again: “Soon I will start a radiological study of K-127; I will let you know.” And then another kind of study was announced. With every one of her messages I became more upset, both at her and at myself for having opened my mouth.

  But I was equally determined to right this wrong. Back in Bangkok, I went to consult with my new friend Eric Dieu, the art dealer who originally set me on my trail by giving me the name of Chamroeun Chhan. He was lounging in his gallery when I entered, studying one of his precious art books, perhaps estimating the true value of one of his priceless statues. “Ah, bonjour Monsieur Aczel,” he said. “How did it go in Cambodia?” I told him about my adventure, about finding the lost K-127, and then I sat down across the desk from him and put my hands over my face. “But I lost it . . . I stupidly, stupidly lost it, by talking about its importance to a Sicilian archaeologist. I think that she must have thought that if it’s that important then she wants it for herself.”

  “Sorry to hear that,” he said. “Now it will likely be sold at auction somewhere in the world and disappear into the hands of some anonymous collector.”

  He sure knew how to cheer someone up. “Do you think there is anything I can do?” I asked.

  “No, I don’t see how. She senses it is important—probably without really understanding what makes it important,” he said. As a northern European, Dieu clearly did not think highly of Sicilians—archaeologists or not. He made a joke about how the Sicilian Mafia now might get its hands on K-127. I didn’t think the situation was as bad as he’d imagined, but I still worried about the artifact and what might happen to it.

  I tried to assess the damage: What was Pellegrino after? Would she really try to sell a rock with an inscription weighing several tons? And how would she get it out of Cambodia? This didn’t make much sense to me; it was an art dealer’s point of view, I understood. It seemed much more likely to me that as a low-level academic (one whose main responsibility was teaching, not research), she was probably interested in making a name for herself—so that she might afterward be promoted—by writing academic papers about K-127 and receiving credit for a find that wasn’t really hers. I had spent too many years as a professor at a small New England college not to be fully aware of the prevalence of such motives in the academic world. But why would she be writing me all these e-mail messages about the artifact? This made little sense to me. Did she, somehow, crave my approval for what she had done and was continuing to do? Weird as it sounded, this seemed to me a reasonable hypothesis.

  After getting another message in which she said she was now waiting for some Italian expert in material science to look at K-127 and determine its structural stability, I lost my patience. Fearing these tests might really endanger the artifact, I wrote to her: “This piece is important for mathematicians and historians of science. Its discovery is the result of my years-long research. Had I not walked into that shed at that particular time when we met, you would know absolutely nothing about K-127 and its importance.” After this, there were no messages from Pellegrino for a week. Casselman wrote me, “Do you think it was wise to write her such a message?” I had to admit that he was probably right. The following week, there was a terse message from Pellegrino: “You are welcome to visit K-127 any time you like. I am here at the lab with the students from 9 to 12 and from 2 to 5, Monday through Friday, from now until May 15.” That was all she wrote. I scratched my head. I was baffled by her responses. Structural analyses were not necessarily noninvasive, and at this point I didn’t trust that she wouldn’t try to “restore” the piece, as she had originally planned. After all, restoration was what her lab, training, and teaching were all about. Worried and disappointed, I flew back home. I never heard a word from Lorella Pellegrino again.

  No, I will not give up, I thought. I will spend every penny I have, if necessary, to save K-127 from being destroyed. I began to plan another trip to Cambodia, with th
e express purpose of meeting with His Excellency Hab Touch of Cambodia’s Ministry of Culture and Fine Arts. I had a long e-mail correspondence with Hab Touch about setting a date to meet in Phnom Penh. We finally settled on dinner on Saturday night, March 23, 2013. I flew to Bangkok a few days earlier and checked into the Hansar Hotel downtown. This was a comfortable hotel with friendly staff, and my large room had space for my rapidly growing collection of working materials, but it certainly was not the Shangri-La, right on the river with its large swimming pool set in a tropical garden. The first thing I did was visit Eric Dieu again.

  I wanted Eric’s advice on how to handle this situation. I found him sitting at his desk in the gallery in a cheerful and relaxed mood. He was elegantly dressed and sported the same expensive gold watch with the movement visible on the face. “What brings you again to Bangkok, professeur?” he asked.

  “I am meeting the Cambodian director general of the Department of Cultural Affairs in a few days, in Phnom Penh,” I told him, “and I hope I can convince him to wrench K-127 from the Sicilian archaeologist and place it in a museum.”

  “Well,” he said, “I wouldn’t hold much hope for that.”

  “Why?”

  “You see, you have to understand how these people think. They have millions of artifacts here, and they just don’t mean much to them. There are storerooms full of statues and stone carvings of every kind below the museum floors. Have you seen the Bangkok museum?” I said that I had, and admitted that it was indeed in a very sad shape, with dirt on the floors and paint chipping from the walls. “If you can,” Eric said, “try to convince the director general to have K-127 sent to a European or American museum—it would be much more appreciated there.”

  I thought about this unpleasant possibility for a moment—I believed that the artifact belonged in Cambodia and not elsewhere—and he went on. “These people,” he continued, “they only care about money. The only one who wants the inscription, apparently, is that archaeologist of yours from Palermo. And even if she wants it for science’s sake—and attaching her name to an important find, as you seem to think—others may easily steal it from her.”

  “How?” I asked. I had thought K-127 was at least safe from leaving the country, especially since it must weigh several tons.

  “Just like that,” he said, snapping his fingers. “They just pack it in a crate, send it by train to Thailand, it is loaded onto a ship, and the next thing you know it is in the hands of some collector in Europe or the US. Don’t be naive about these things, professeur . . . It’s pretty easy to smuggle any kind of artifact, especially when a lot of money is involved—and from your description, this stele could well be worth many millions of dollars because of its historical significance. And, you never know . . . It could be gone already, lost forever.”

  I told him that it was supposedly still in Siem Reap, since Pellegrino’s message said that I could come to see it again any time I wanted. At that moment, a tall blonde woman walked in, and came over to Eric’s side. “Votre femme?” I said. “Madame Dieu?” He smiled and nodded, and I introduced myself to her. I now felt it was time for me to leave. “I’ll try my best,” I told him.

  “I hope you save the artifact,” he said. “It is a noble cause. I applaud your trying to save it for science, for history . . .”

  24

  In the morning of March 23, 2013, I took the train to Bangkok’s Suvarnabhumi International Airport and waited to board a Bangkok Airlines flight to Phnom Penh. I pondered the fate of the artifact I was trying to save: Was it still in Cambodia? Could it be saved? Again, I blamed myself: I would not be in this predicament if I hadn’t been so talkative.

  Some people fish or do crossword puzzles to relax; I think about prime numbers. An inscription so important to the development of numbers sure had an auspicious number itself, 127. It was not only a prime number, but it was also one of the Mersenne prime numbers. Mersenne was a Minim monk living in a monastery in Paris and a close friend of René Descartes. Both loved numbers and mathematics. Mersenne was Descartes’s correspondent while the philosopher traveled throughout Europe and was often the only person who knew Descartes’s whereabouts. Descartes was secretive and worried about persecution by the Inquisition because of his embrace of the heliocentric theory of Copernicus, which the Catholic Church abhorred. This was about the time of the trial of Galileo, 1633, and many other thinkers and intellectuals worried about their views of nature becoming known to an unfriendly and very powerful religious establishment.

  In some of their letters, Descartes and Mersenne discussed numbers. Mersenne became convinced that numbers of the form 2p – 1, where p was a prime number, were always prime. Such numbers became known as Mersenne primes. Let’s see how it works. The first prime number is 2. So the first Mersenne number is 22 – 1 = 3, which is, indeed, a prime number. So what is the second Mersenne number? Since the second prime number is 3, the second Mersenne number must be: 23 – 1 = 7, which, again, is a prime number. What comes next? 25 – 1 = 31, also a prime. Next comes: 27 – 1 = 127, and indeed this number also happens to be a prime number. So 127 is not only a prime number—it is also the fourth Mersenne prime. It is thus a very special kind of number.

  Mersenne thought that what he had was a theorem, meaning that it was always true that numbers of this kind are primes. He could not prove this theorem, and in fact it turned out to be false. In fact, the very next Mersenne number is not a prime, because: 211 – 1 = 2047, which is the product of 23 and 89. (But in general, Mersenne numbers are frequently primes.) This makes 127 even more special, I thought, since it is the last Mersenne prime before we reach the first nonprime Mersenne number. I now felt absolutely ready to try to retrieve K-127 for science history, and at that moment my flight was called and I walked fast to the gate.

  Standing in line to board the airplane, I remembered the best story I know about a theorem that turned out not to be a theorem at all—as in the case of Mersenne’s idea on prime numbers.

  The Great Internet Mersenne Prime Search

  While Mersenne numbers are not always prime, they do provide a very good way of searching for large prime numbers. All one has to do is plug in the largest known prime number, p, to produce the number 2p – 1, since this number is a good candidate for being prime. And it will be exponentially larger than p. One can then check, using mathematical routines, whether this new number is indeed prime. A program called GIMPS—the Great Internet Mersenne Prime Search, in which distributed computing worldwide is being used to search for ever larger prime numbers—has resulted in the finding that 257,885,161 – 1 is a prime number. At this writing, it is the largest known prime.

  The well-known Japanese American mathematician Shizuo Kakutani was in Europe right after World War II visiting another mathematician. They took a walk together in the German countryside and discussed mathematics. Now, right after the war, US forces had a large presence everywhere in Germany, with military bases and camps located throughout the western part of the land—the regions that had just been liberated by the Allies. Eventually, the two mathematicians, deep in a discussion, inadvertently walked onto an American military base. The guard at the entrance to the camp ran after them, furiously yelling at them to stop. “What are you doing?” he demanded. “We are talking,” answered Kakutani. “Talking about what?” shouted the guard. “We are talking about a theorem,” answered Kakutani. “What theorem?” the guard demanded. “It doesn’t matter,” answered Kakutani. “It turned out to be false.”1

  As I stood in line, still waiting to board the Airbus A319 jet on flight PG 933, I closed my eyes and realized that K-127’s inscription date, 683 CE, was also a prime number. I could not tell how I knew it—it was an intuition. Later, sitting in the airplane, I made a few lengthy mental calculations (I did not bring a calculator and didn’t even have pen and paper, so I had to do it all in my head) and verified that 683 was indeed prime. Everythin
g about this search had an almost supernatural mathematical flavor. I hoped the present stage of my quest would be a success, and the fact that 683 was a prime number made me feel that I had found an auspicious sign.

  In an e-mail message before I left Bangkok, Casselman had expressed skepticism. He noted that as far as he knew, Cambodia was a country in which corruption was rampant, especially in officialdom, and that I might not get anywhere. I told him about my experience in Laos and added that I was ready to face anything that might happen. A woman sitting next to me on the plane, a Canadian who worked for an international organization in Phnom Penh dedicated to getting women and girls off the streets and into reasonably paying jobs—perhaps the organization to which Nicholas Kristof had taken the two girls—told me some horror stories about her neighbors being threatened with deportation for an overextended visa unless several hundred dollars were paid as a bribe. Someone I told about my trip had asked me if I was willing to bribe an official in order to save K-127. I said absolutely not. Besides, I had a very good feeling about His Excellency Hab Touch, my government connection in the quest to save the inscription.

  25

  I knew that I was dealing with a very modest man when I called His Excellency Hab Touch from the InterContinental Hotel and he said that he would meet me there and would call my room upon arrival. I said, “No, please let me wait for you downstairs.” But he refused: “No, no, you relax in your room and I will call you when I arrive.”

  Our meeting was set for 6:30 p.m.—I had half a day. I sat in my room, high on the ninth floor of the hotel, and looked out the window. The contrast between Phnom Penh and Bangkok was stark. In the Thai capital, every building is shining, bright, and painted white or light blue. They all look like they’ve just been built; you never see water stains running off windows, the streets are so clean you could walk barefoot on them, and everyone smiles. Thailand is a prosperous land where tourists—a mainstay of the economy—are treated as royal guests. Despite the political problems the country has endured, including martial law, most Thai are immensely proud of their king and his family, and the royal family’s pictures adorn many public places.