Numerals-of-Ahaxa--350The Numerals of Ahaxa


Robert Dawson


Along this stream, time moved as slowly as the muddy water. The insects, the trees, and the steamy jungle reek of the Amazon Basin were as they had been before humans came here. Bushes smothered the hidden banks, stretching out low over the water for precious space and sunlight, and the slow greasy current slid under and through them. Monkeys swung through the linked branches overhead, crossing from one side to the other without slowing. Strange birds hooted in the distance.

Only the inflatable boat, its cargo, and its taciturn pilot were of the twenty-first century. I watched the boat’s wake disappear under the bushes, tried not to think of snakes, jaguars, or Apocalypse Now, and wondered, for the hundredth time, why anybody would need a mathematician – a number theorist – in a place like this.

The boat rounded one more corner. Fifty meters ahead, a crude log dock jutted into the lazy current. Was this, finally, Ahaxa?

As we drew nearer, natives appeared out of the forest in ones and twos, watching with silent curiosity. They wore loincloths of pounded bark, ochre stripes on cheeks and forehead, and brightly-colored feathers in their hair. And – yes! – there was Dr Fernando Costa da Silva, whose urgent phone call yesterday had brought me here from my sabbatical in Rio de Janiero.

He had not changed much in ten years, though his clothing – boots, khaki shorts with many pockets, a scruffy shirt, and some sort of hat – was rather more informal than what I remembered from his student days. Beside him, a striking dark-haired woman in her forties wore capris, a short-sleeved shirt, and a malachite-green feather in her hair as a gesture to the local fashion. The pilot killed the motor and we drifted silently to the dock.

“Professor MacLean! Bem-vindo ao Ahaxa!” Fernando said, as he helped me out of the boat. His headgear proved to be a battered baseball cap advertising the sports teams of my university. “Have you had a good trip?”

It had only been a little over twenty-four hours since Fernando had insisted that my expertise was needed in Ahaxa, and that plane tickets were waiting for me at the airport – words calculated to appeal to the inner Indiana Jones of any middle-aged academic. Since then I had been travelling almost every waking hour: a commercial plane to Manaus, then a private floatplane, and finally the little inflatable boat. I was utterly tired. “It was all right.”

“This is my boss, Doctora Juliana Rosário. Juliana, this is Professor Michael MacLean.”

“’Mike’, please! So, now I’m here, what’s all this about?”

“It’s an ethnomathematics thing. You’ll love it!  Listen to how they count here: ya, ki, deh, kiki, mor,…”

We were interrupted by the pilot, saying something I could not catch. “He says he has to go back,” Fernando explained. “We must unload now.” So for five minutes, under the pilot’s dignified supervision, we unloaded plastic crates and heavy drums of liquid – methanol, Fernando said, for the fuel cell generator.

“So, what’s so special about the way they count?” I asked, as I heaved another drum of methanol out of the boat.

Fernando laughed. “It’ll be easier to explain when I can show you – but you’re going to be amazed!”

He passed the drum down to Juliana, who looked at us severely. “I don’t know what Fernando is telling you, but it is not just about the numbers on the stones. You are not here to play. I found the money to bring you here because this village, Ahaxa, is in very serious danger.”

 “Well, mathematics is my line. Saving villages I don’t know much about,” I said.

 Juliana said something to Fernando that I could not follow, but from the toss of her head in my direction and the words matemático inútil – useless mathematician! – I gathered that she was not entirely impressed with me.

We finished unloading. The boat headed back downriver, leaving a faint trail of blue smoke hanging in the still air. Juliana lifted a fifteen-liter fuel drum to her shoulder and stalked off.

Fernando hoisted another; I grabbed my suitcase and laptop, wondered if I could manage another item, to be less inútil, and decided that I couldn’t.  Some of the watching Ahaxans picked up containers too, and Fernando led us into the village.          

My first impression of Ahaxa was a maze of thatched huts, huge looming trees with buttressed trunks, stone slabs taller than a man, vegetable patches, and animal pens, all mixed at random; I could not guess how far it all stretched. Streets were everywhere and nowhere, a spider-web of tracks going around and between everything. Villagers looked up from their work and nodded as we passed; small children pointed at me and giggled, older children shushed them.

I caught up with Fernando. “Fernando!” I said in an undertone. “What’s up with Doctora Rosário?”

“Don’t mind her. She’s worried, that’s all. Worried about the village.”

 We walked for several hundred more meters and finally caught up with Juliana near a small cluster of huts marked by a hand-written sign: ‘Estação de Pesquisa Antropológica do Ahaxa’.

“Welcome to the research station,” said Fernando. We put down our loads, and he went to each Ahaxan in turn, clapping his hands firmly onto his or her shoulders, and repeating some formal phrase of thanks. They scattered back among the huts.

“Should we go back for the rest?” I asked Juliana.

“No. This is Ahaxa. Nobody steals anything here.” She sounded affronted. “We’ll get it later.”

We ate dinner in the open, crowded around a small folding table. The food was rice and grilled fish steaks. “This smells delicious!” I said. “What is it?”

“It is pirarucu” said Juliana. “A very big fish. Only the natives are allowed to catch it.”

I hesitated. “I’ve heard of it. Isn’t it endangered?”

Em perigo? In many places, yes, but not here.” She smiled. “The Ahaxans are careful fishers. You may eat it.”  I did; it was as good as it smelled, and I was hungrier than I realized. Once the edge was off my hunger, I asked Fernando again why I was there. He launched into his exposition, gesturing with his fork to emphasize points.

“Well, you have to realize just how special this place is. Most of us linguists believe that writing has only been invented twice, once in Mesopotamia and once in Central America, with everybody else copying the idea from their neighbors. I’m becoming convinced that Ahaxa makes a third.” He pulled a folder out from under his chair and showed me photographs of slabs like the ones I’d seen on the way in, covered in inscribed patterns. “These incised slabs are called kaximeroxata in Ahaxan. They’re all over the place, in the village and out in the forest. We’ve found hundreds of them already.  They’re all ancient.  The natives keep the important ones cleaned, and some of them can read the easier inscriptions, but nobody makes new ones anymore.”

I swallowed a mouthful of pirarucu. “That’s definitely strange. But how do I come into it?”

“Most cultures’ early written records are history or arithmetic. Well, the history might be myth, and the arithmetic is often accounting – but those were the first things people thought to record. Here we haven’t found anything that looks like history yet; in fact, the only glyphs I’ve been able to give definite translations for are numerals.”

“A people after my own heart; I never liked history classes either. But I still don’t see why you need me here.”

“Well, here’s how you count to ten in Ahaxan: ya, ki, deh, kiki, mor, kideh, shu, kikiki, dehdeh, kimor.” He reached down beside his chair and picked up a printout. “Here’s how they look written out.” He waited expectantly, as if he had asked me a riddle.

I looked. Suddenly it hit me. “Good God! Prime factorization! Two, three, twotwo, five, twothree, seven, twotwotwo…”

“That was fast! It took me weeks to figure that out, when I was first learning the language.”

“I suppose this wasn’t their original number system.”

“Of course not, any more than our culture has always had base ten positional notation. You taught me that! But there’s no trace of anything earlier now, either written or spoken.”

After dinner Fernando took the dishes to wash, leaving me with Juliana. We sat and talked by the pale light of a little LED lantern that cast dark shadows across her tanned face.

Fernando says he studied with you in Canada. Do you also teach linguistics?” she asked.

“No, but he studied linguistics at my university. He took my Liberal Arts Math course and was one of the best students. I got him to take a few more courses after that, and he ended up doing a minor in math.  I tried to get him to change his major, but he stayed faithful to his first love.”

She smiled. “I see. And is he right, that there is real mathematics on the kaximeroxata?”

“It looks like it, though I don’t understand it yet.”

“But we’ve been here for over a year, and we haven’t seen anything that they use mathematics for.”

“Well,” I said, “think of Ancient Greece. They didn’t have much technology, and apart from Archimedes, who was his own breed of cat, the free citizens who did the mathematics didn’t even use what they had. Craftsmanship was left to slaves.” I thought for a little. “And there was the Japanese temple geometry in the Edo period.”

“What was that?” she asked.

“They would prove really difficult geometry theorems, then paint them on boards and hang them up in temples instead of publishing them.”

“Oh, Mike! Don’t be so eurocêntrico!  That was publishing too, wasn’t it?”  Her laugh was low and intimate.

“Yes, I suppose it was.”

Fernando reappeared with cans of almost-cool beer. There were three whispered explosions, and we drank gratefully. Somewhere in the village a woman was singing, a high keening song in an unfamiliar scale. In the darkness behind us frogs croaked. Life was good.

“So, Juliana,” I finally asked, “you were saying something about this village being in danger.”

She frowned. “Sim. See that tree there?” She pointed to a huge trunk that disappeared into the darkness above us like a fluted cathedral pillar.  “That’s jacarandá…” She looked at Fernando.


Sim, ‘rosewood’. That tree is hundreds of years old.  There are not many rosewood trees this size any more.  This one might be worth a hundred thousand reais. More, maybe.  If loggers come it will destroy the village. Their food, their way of life. Everything.”

“But this site is unique!” I said. “They can’t do that!”

Juliana looked at me and rolled her eyes. “A grana fala mais alto.” Money talks louder. “People know about Ahaxa. Maybe it is too late. But perhaps if the mathematics is very important, that will help.”

The song ended; the frogs continued tirelessly. It had been a long day; once my beer was finished, I excused myself, went to bed, and slept immediately, despite the sticky heat, the jungle noises, and my curiosity. Even despite Juliana.


Next morning, I was the last one to breakfast. Coffee was ready, the traditional cafezinho made in a little saucepan. I filled myself one of the tiny cups and sipped slowly.  Juliana was saying something in Portuguese about logging permits; her face was like a thunderstorm.

“What’s wrong, Juliana?” I asked.

She looked disgusted, and waved at the boxy satellite phone on the table. “We have just heard from Brasília that IBAMA, the environmental agency, is maybe changing its mind soon about protecting this area from logging. Like I said they would.”

“You worry too much,said Fernando. She picked up the phone, and left, not deigning to answer.  I thought about going after her, but Fernando opened a folder and started to explain to me what he knew of the Ahaxan number system.

I nibbled on a piece of a blue-skinned fruit, a little like a mango but more tart, and tried to pay attention, though my mind was on Juliana. After a while I stopped him and asked: “Tell me – how high do they go?”

“Now? Most villagers can’t count to a hundred. How often does a hunter need to know the root word for ninety-seven?  If they have to, they use words like `kikimormor’ to mean a really big number, even if it’s indefinite.”

“`Kikimormor,’ a hundred. Interesting; a hint of the base-ten concept. I wonder if that’s a more modern development?  But what I really meant was, how high do the prime number glyphs go?”

“That’s the other odd thing: we don’t know. All the prime number glyphs – and no others – have that wavy underline there. Believe it or not, some inscriptions have glyphs for hundreds of different primes, some very complicated.  But there’s no obvious pattern to them, any more than knowing um, dois, três tells you what quatro means. Nobody knows how to read the higher ones.  Maybe you can help.”

“Well, did they seem to be in a list or table?”

“No, little clusters, with other symbols, like equations.” He showed me a list of glyphs without the wavy line. “Here are some of the common ones that I’ve found with them. I think the first is a plus sign. If it is, then this one means one hundred and ninety-nine.” He pointed to a glyph with about ten strokes.

I thought for a bit. “If they aren’t just a list, then they probably aren’t consecutive; so if you have, say, five hundred glyphs, you probably have a sample, with gaps, from the first thousand primes or more.  So some of those could represent primes as large as ten thousand, or bigger.”

Meu deus! How does anybody work with those, Mike?”

“I wish I could tell you, Fernando. All I can say is, I’d want a very good computer.” I pointed at a shape like a squared-off infinity sign with one glyph in each loop; similar patterns were repeated elsewhere. “What’s that?”

“That? We call that the frame notation. For some reason, early inscriptions often use this convention, but for even numbers only. The frame contains two prime glyphs, always two, added rather than multiplied. So six would be deh-xa-deh, eight deh-xa-mor.”

I knocked my coffee over. “Goldbach’s conjecture!”

“What, Mike?”

“You can only do that if Goldbach’s conjecture is true. It’s one of the oldest unsolved problems in mathematics. Goldbach was a contemporary of Euler, about two hundred and fifty years ago. He wrote to Euler and conjectured that every even number except two is the sum of two primes. It’s almost certainly true, but after all this time, it’s still unproved.”

Fernando was thoughtful. “I think it’s time we should talk to my colleague Lexofiti. She knows more about the kaximeroxata, the number stones, than anybody else in the tribe.  Maybe she can tell us more.” We got up and walked through the village.

 We were passing a wooden pen, when I heard a chittering sound like a baritone squirrel. I looked inside; it was occupied by three animals, half a meter high at the shoulder, that resembled little furry hippopotami.  They looked up at me curiously. “What the hell?” I asked.

He smiled. “Capybaras. The Ahaxans are mostly hunter-gatherers, but as you see they do some farming. They have an unusual system of crop rotation, and some crop plants that they seem to have bred themselves, quite different from the wild strains. The agronomist who was out here last month was quite excited about it.”

We walked on, and eventually came to a thatched hut like hundreds of others. Beside it, a lean Ahaxan woman with iron-grey hair was weeding a row of vegetables with a wooden hoe. Fernando greeted her. “Hé, Lexofiti, lahaminu!”

They conversed in machine-gun Ahaxan, occasionally throwing in a scrap of Portuguese. After a while Lexofiti gave me a toothy grin, put down her hoe, and led us through the village to a particular slab. She spoke some more to Fernando, and ran her finger along a line of inscription.

It contained, prominently among a long string of other symbols, a six-factor number. The first glyphs were those for two, seven, and –- I thought for a moment — seventeen; the next, repeated twice, was a complicated crisscross of about twenty strokes.  The fifth was of unimaginable complexity, like a street map of an entire medieval town.

Fernando turned to me. “Lexofiti says her ancestors stopped using the frame notation because they found it was no good. They found that this number could not be written that way.”

The birds screamed. I felt lightheaded, and put my hand on a nearby tree to steady myself. Finally I spoke. “Fernando. I need to get this straight. Do you realize how incredible this is? Is Lexofiti really telling us that this number here is a counterexample to Goldbach’s conjecture?”

They talked again, for several minutes. All I could follow was that Fernando seemed to be asking questions, Lexofiti answering and pointing out the relevant bits of the inscription. Sometimes I thought Fernando repeated himself, and then Lexofiti’s response seemed just a little impatient. Once she turned and pointed at me, grinned and said something. Finally Fernando explained. 

“Yes, she says. She is quite definite, but it was very long ago. She says that this, that our ancestors thought about this too, this makes our tribe sisters of her tribe. But she does not know how they did it.”

“Fernando! Please! Ask her what that number is, and that one.” I indicated the repeated twenty-stroke glyph, and the enormous one – sweet holy Jesus, how big would a prime have to be to need that complicated a pattern to label it? Maybe, just maybe, the product of the six primes was small enough that a modern computer could check by brute force whether it was a sum of two primes?

Fernando said something in Ahaxan. Lexofiti answered, ending with an expressive shrug. Fernando turned to me, apologetically.

“She says ‘Who knows? It is very old. We do not use numbers that large anymore.’”

Somewhere in the trees, a bird hooted derisively.


We went back to the compound and returned to work. By lunchtime, we had tentatively identified glyphs for multiplication and subtraction, and had identified all of the number glyphs up to a hundred. Then, mid-afternoon, we had a breakthrough.

“Hey, Fernando,” I said. “Look! I think these are Pythagorean triples! See, three-four-five, and I think that’s five-twelve-thirteen? Real mathematics! But I can’t make out this bit of the transcription – we need to check the original. Where is it?”

“Let’s see.” He consulted a sheet. “Number eighteen? That one’s down by the dock. You passed it yesterday on your way in. Let’s go take a look.”

We jogged through the village, our glyph list in hand, and spent some time standing in front of the kaximeroxata, translating it all over again, like a pair of tourists with a guidebook. Finally we agreed that we really had it right, and that there was nothing else that it could mean. We exchanged high-fives, and went for dinner.

The meal was awkward; Fernando and I were in a mood to celebrate, despite the tantalizing setback of the morning; but Juliana was unhappy. One angry sentence at a time, the story came out. There had been another phone call; somebody high up in IBAMA had decided to review the ban on commercial activity around Ahaxa in early March.

“But that’s only five weeks from now!” said Fernando.

“That’s right,” Juliana said. “I do not think the loggers want to wait.”

“But this place ought to be a United Nations World Heritage Site!” I said.

“It will take more than a few scratches on rocks to make that happen!”

“But we have more!” He explained what we’d learned; she listened, unconvinced.

“So they proved this Goldbach conjecture?” she asked.

“No,” I said. “We think they showed it wasn’t true.”

“Is that important?”

“Important? This problem’s been open since… since Brazil was a kingdom! If I’d discovered this myself, I’d be in line for one of the big international awards!”

“Like the Fields Medal?” asked Juliana.

“You know about that? No, I’m afraid I’m a few years too old; you have to be under forty. But still, if we knew that Lexofiti was right, it’d be really, really big.”

“You said `if’? Do you think she is wrong?”

“I don’t know,” I admitted. “I’d really like to believe it. But even if one of the big names in my field – Laforce, say, or Yermakova – said they had a counterexample, we’d expect to see it. And nobody knows what that number is anymore; and from the number of strokes in that last glyph, I don’t think we could find it again in five weeks on any computer that I can afford time on.”

“What about these Pythagorean triples you mentioned earlier? Are they new?”

“No. The Babylonians knew about them four thousand years ago. Lots of other cultures did too, since then.”

She sighed. “Then we still have nothing.” She squeezed my arm. “But thank you for trying, Mike.”

I put my hand on top of hers. “I haven’t given up yet.”


The next day, at dinner, Juliana asked: “About how much computer time would it take to find that number?”

“I don’t know. We know the small factors; that helps a little. We know there’s a repeated factor; that helps too. We can guess about how big the factors are. Maybe a year on a supercomputer, maybe more.”

“Good! Then I think I know how we can do it. My sister likes to read – ficçião cientifica. How do you say that?”

“Science fiction.”

“ When she does not use her computer, it runs a program that checks radio telescope signals for messages from other stars. There are many, many people and computers involved; she says, together, they are more powerful than a supercomputer.”

I thought about the implications of this, as the birds squawked their commentary in the trees. Finally, I said slowly; “That’s a really good idea.  There are projects like that in mathematics, too, things like the Mersenne Prime Search. But the networks take time to set up. People aren’t going to turn over their computers to any crackpot who comes along with a story about lost secrets of the ancients.”

“Cracked pot?” Juliana asked.

Maluco,” explained Fernando.

“Then we must show that we are not cracked pots. You have found one thing already, Mike? Maybe you can find more?”

“I’ll do my best,” I said. “But you’ll both have to help. And before we start, I think it’s probably my turn to wash the dishes.”


As I washed, I thought about what we could do; and that evening, we laid out our strategy. Fernando and I listed mathematical ideas that we might recognize from distinctive number patterns: the Fibonacci sequence, multigrades, Paley sequences, approximations to pi. We made up lists of glyph sequences to watch out for. Then, too tired to do any more, we slept.

Next morning, Fernando and I sought out Lexofiti, while Juliana went to find the other village elders and enlist their help. We found Lexofiti squatting on the packed earth beside her hut, twisting a length of thin cord out of some sort of light-brown fiber. Silent and impassive, she listened to Fernando’s explanation; then put down her work, and rose slowly to her feet.

We followed her to a hut, across the village. Juliana was standing a few meters away from the door, looking nervous; a heated argument was taking place inside. Lexofiti walked past her, into the shadows inside. Her voice joined the others. Maybe half an hour later, she and four other elders emerged. One of them spoke to us. Lexofiti added a sharp staccato comment.

 Fernando listened diffidently, said something in reply, and turned to us. “They are not happy about this.  Lexofiti says we are playing with fire – those are her words, melahata nunu taga. But they agree– Lexofiti too – that it is the best chance they have, and they will assist us.”


By noon Ahaxa was in the grip of a village-wide scavenger hunt. Children ran around the village clutching scraps of paper, trying to find matching inscriptions. Adult Ahaxans faded, jaguar-silent, into the jungle, searching out rarely-visited kaximeroxata in places no longer inhabited. Juliana wandered around with a camera and notepad, coordinating us all. Lexofiti, Fernando and I sat in our improvised mission control center, adding to our rapidly-increasing list of glyphs and building up translations of inscriptions.  The Ahaxan Number Theory Workshop was under way.

Two days later, our dictionary of Ahaxan mathematical symbols had grown enormously. We had symbols for fractions, powers, factorials, and even something that seemed to be Euler’s phi function. We had notation to define and apply new functions. Perhaps the biggest breakthrough came when we realized that one set of simple glyphs were variables – like x,y, and z – meaning different things in different places.

 And we had more than twenty inscriptions interpreted, ranging from simple identities among the Fibonacci numbers to some subtle results in combinatorics.  Speaking through Fernando, I thanked our many research assistants, and we called the hunt off.

Lexofiti joined us for dinner. Fernando had prepared a meal of stewed capybara; the flavor was a little like rabbit. For a while we ate without speaking. Finally Juliana asked: “What happens next?”

“First, I’m going to write this up – including the threat to the village. Then, if I can borrow the satellite phone again, I’m going to connect it to my laptop and send the result off to Dr. Baez at Riverside. His blog is practically required reading in the math world. By tomorrow evening this will be everywhere.

“I’ve already contacted one of the people involved in the Mersenne prime project, and he says he can have the Goldbach code ready by then. And with any luck, people will be ready to sign on.”

Juliana turned to Lexofiti and asked her something. Lexofiti frowned and was silent. After a minute Juliana spoke again, and this time Lexofiti answered, though her voice sounded reluctant. Juliana turned back to me.

“Mike, I asked Lexofiti whether it was all right to include pictures of the kaximeroxata. She told me they are for everybody to see, that is why they are all around the village.  So why don’t you include some of the ones you haven’t solved yet?  That should get people interested.”

My first reaction was to say no, this was our project. But I had to admit that she was right; it was the perfect hook.  “OK, we’ll give them that one with all the square roots to start with, and maybe the one that looks as if it might be a double-counting formula.”

“You can decide which ones. But I think it will get a lot of people interested.”


By the morning of the third day, eight hundred computers around the world were running the Goldbach code, with more linking in every hour. The one we’d thought was a double-counting formula had been decoded, and somebody had found it in a graph theory paper from the 1980’s; the other was still unsolved and had its own wiki.  I sent out two more tough-looking inscriptions, and worked with Fernando on another.

Juliana brought lunch to us at the table where we were working. “Fernando?” she said. “The fuel cell generator has stopped again. I could not make it work. Can you try please?”

“I suppose so. Mike, can you carry on without me for an hour or two?”

“I’m not sure. I think I need to check the original of this inscription here.” I pointed to one of the photographs.

“Which one is it?” Juliana asked. She checked the index number on the sheet, and the master list. “That is one of the new ones, isn’t it? I remember where that one is, Mike. I will take you there. Sem problema!

After lunch, Fernando went off to the kitchen hut, and Juliana and I went off to find the kaximeroxata. The path was just wide enough for two people to walk side by side. As we walked, we talked about this and that; the midday heat, and other things, made it hard to concentrate on more serious matters.

We reached the kaximeroxata and I checked the inscription; it was not quite what I’d hoped it would be, but I made a careful sketch of the glyphs anyway.  On the way back, Juliana stumbled a little, and our arms brushed. Almost instinctively, I reached to take her hand, but she pulled away. “No, Mike.”

“I’m sorry. I didn’t mean anything by it. For all I know you’re married or something.”

She was silent for a while. When she spoke, her voice was hardly more than a whisper. “No, Mike, I’m not. And I do like you, maybe too much.  But you are going away in a few days, and –I don’t want to get hurt. Please, let us be sensible?”

With a tight little smile, she turned and started back, keeping to the middle of the path. I followed behind, trying not to torment myself by gazing at her as she walked along the trail – jungle-scratched legs, beat-up hiking boots, and all – but it was impossible. A bird hooted. I swore under my breath and kicked at a stone.


During the next week, the number of computers working on the Goldbach Project passed ten thousand: not as large as the really big distributed-computing projects like SETI@home, but pretty respectable. The media were getting onto the story. Newsblogs around the world ran daily updates on the Goldbach Project, and printed our latest kaximeroxata transcriptions as if they were chess problems. I must have had to do a hundred interviews; at least it kept my mind off Juliana.

Finally I managed to find the satellite phone unoccupied and made a call to the Dean of Science. His secretary put me through immediately.

“Hi, Mike. So it really is you behind this, is it?”

“Yes, Peter.”

“Rio wasn’t exciting enough for you?”

“It’s a long story.”

“I can wait to hear it till you get back, but Public Affairs will kill us both if you don’t get something to them immediately. About two thousand words? And a couple good photos they can use in the press release. OK?”

“Sure thing. In my copious free time.”

“Tell me – are you close to solving the problem?”

“People are excited about it, that’s what counts. By the way, there’s something I need from you – just in case this works out – think you can help me?” I explained what I had in mind.

“Well, we’ve got a Senate meeting coming up on Friday. I guess I can get that onto the agenda.”

“Er, Peter, what day is it today?”

“It’s Wednesday.” He chuckled. “You do realize you have just confirmed all my worst stereotypes about mathematicians, don’t you?”


Days passed. I was pacing the ground gloomily in front of a kaximeroxata inscription, when Juliana came running from the direction of the research station.  She threw her arms around me and almost knocked me off my feet.“Nós ganhamos, Mike! Nós ganhamos!”


 “We have won! IBAMA have said there will be no logging!”

I grinned. “You mean they’ve read the writing on the wall and told the loggers to go forth and multiply?”

She looked puzzled. “If that is a joke, you will have to explain it for me.”

“Never mind. It wasn’t that funny. So what’s happened exactly?”

Without letting go of me, she told me the good news. A fifty thousand hectare region around Axaha was going to be declared a national park with very limited access, and, yes, an application was in for UN World Heritage status.

“But for a year there will be people coming from the Nações Unidas to decide about this. So, Mike, you will have to stay with us, to explain the mathematics to them. Can you?” She looked up at me pleadingly.

The university had made it very plain that they wanted as much publicity out of this as I could manage. Did that extend to a year’s extra leave, I wondered? “Yes. I can,” I said, hoping it was true.

She smiled, closed her eyes and raised her lips toward mine. For a few minutes the frogs and birds had the floor to themselves. After a while she whispered: “I think we should stop now. There is still a lot to do today.”


Half an hour later, the three of us wound our way through Ahaxa to Lexofiti’s hut. With Fernando translating, I laid out what the Senate had agreed to.

“Remember when you said that my tribe and your tribe were sisters? I have asked the elders of my own tribe, and they say that I can stay here, even after the United Nations elders have left, to teach you, and any other Ahaxans who want to learn, about the things your ancestors knew, so you will know what the kaximeroxata say.”

She smiled at me as an indulgent grandmother might smile at a child who has offered to fix her car, and gestured towards the door of her hut. Slowly, she hauled herself to her feet and led us out, and through the village to where a thin, overgrown trail that I had not noticed before led into the jungle.  We made our way for half a kilometer over rough ground, keeping pace with Lexofiti’s slow but steady footsteps.

Finally we came to a heap of rubble, no piece larger than a golf ball. Lexofiti picked up a piece, looked at it, put it down, and tried another. When she reached the fourth piece she nodded, tight-lipped, and handed it to me. One surface bore a few scratches that I somehow knew had once been part of a glyph.  She talked for a few minutes with Fernando, in an undertone; when she had finished, he translated.

“Lexofiti says that her ancestors went too far, that they learned things that were not good and harmed Ahaxa. People were ill and hungry. So they smashed the bad kaximeroxata and did not make any more. She says nobody knows what was on those kaximeroxata; it is better that way.  She says they honor their ancestors but do not want to repeat their mistakes. They do not want to walk on that trail again.”

“Fernando! Tell her…” Then I paused. Tell her what? Tell her all knowledge is good? That we’ve never been harmed by any of our mathematical knowledge? That people in the developed world are happier than the Ahaxans?  “Tell her that I don’t know if she is right, but I honor her people’s decision. Tell her that if they ever change their minds, we will share what we have learned with them.”

I felt Juliana’s fingers slipping into mine.


It’s been eight months now, and the computers are still plugging away out there. The answer could come tomorrow, or it might be centuries; the current best guess is a year or two.  But the Goldbach Project participants don’t seem to mind; we still have about three-quarters as many computers working on it as we did at our peak. And the only problem we’re having with the UN site visitors is getting them to leave when they’re through so we can get on with our work.

I still hear that pesky bird hooting at us sometimes. But these days I just go over to the door and hoot right back.


Like the protagonist of this story, Robert Dawson teaches mathematics at a Nova Scotian university. His fiction has appeared in Nature Futures, AE, Compelling SF, and numerous other periodicals and anthologies. This is his second story in Aurora Wolf. When not teaching, doing research, or writing, he enjoys fencing, cycling, and cooking. He’s an alumnus of Cambridge, Sage Hill, and Viable Paradise – all weird and wonderful institutions. His fiction page is at