Her Prime Conjecture1

In Issue 75 by Carsten ten Brink

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Photo by BillionPhotos.com on Adobe Stock

‘Lasagne. It’s already in the oven, Mom,’ Cissy said. ‘And then I have a lot of papers to mark tonight.’ One and a half lies in that answer, but they were only white lies.

‘Don’t eat too much of it, honey,’ her mother said. ‘I know how rich your lasagne is. You can freeze the rest.’

‘Yes, Mom.’

‘Your father had a good week. The Chess Team reached the Third Round.’

‘Uh-huh.’

‘I’ll put him on. I’m sure he’ll want to tell you all about it,’ her mother said.

‘You’ve just told her all she needs to know, dear,’ Cissy heard her father say in the background.

‘Dad, I’m proud of you,’ Cissy said. ‘Good luck with the next round.’

‘He heard you,’ her mother said. ‘Did I tell you about the Johnstons? They’ve been having such problems with their roses that Elizabeth wants them to dig them all up, replace them with rhododendrons or something.’

‘Mom.’

‘I know. You don’t care about gardening. But imagine! All those years of trying and now she wants to give up. I don’t—’

‘Mom, sorry to interrupt,’ Cissy said. ‘I need to get the lasagne out of the oven.’ Another white lie. But Mom’s gardening gossip could easily cost her a further fifteen minutes.

‘I understand, honey,’ her mother said. ‘I’d better let you go.’

‘Love you both. Bye, Mom. Bye, Dad. Speak next week,’ Cissy said and hung up her phone.

The delicious smell of warm cheese greeted her in the kitchen. Her lasagne bubbled but was still a good ten minutes from being ready.

White lies, that was all. Two full ones and two half-lies in that conversation.2 She didn’t spend nearly as many evening hours marking Social Science student papers as she’d claimed. And her parents had accepted without question her excuses when she’d said she couldn’t visit for the Fussgoedel family Thanksgiving. They wouldn’t understand if they knew.

It hadn’t been a total lie because she had actually brought student work home, but the task hadn’t taken long. Statistical analysis was a lot easier for her than the wordy arguments of most student assignments. Cissy hated those long, dense paragraphs of confused thinking, odd vocabulary and random footnotes, written by students without an ounce of feel for the elegance and logic of a mathematical proof. But she’d been lucky tonight – her skill with numbers and formulae was unusual in her faculty, and she’d traded review work with her Ellie.3 It was promising to be a good evening.

Cissy laid the table and returned to her bedroom, where she organised her desk. She gave the student papers one last look and returned them to their folder. There were two satisfying thwips as she fastened the elastic bands around the corners.

Five minutes since their conversation and Mom hadn’t called back with an ‘I forgot...’ so Cissy’s evening would be spared any further family distractions.

She unlocked the cabinet over her desk, pulled out her red journal and marked the date onto a fresh page.

Today was the month’s third Thursday and her single friends would be ten-pin bowling. Except Ellie and Petra, who’d head to the Blue Island, to the table they always took, the one Sam held for them. Ellie would have her G&T and Petra her vodka, and because they had to work tomorrow, they’d stretch the drinks out and refill them with soda water, which Sam wouldn’t charge them for. One glimpse of Petra through the window was usually enough to convince men to enter, grab a free table near her and buy drinks. Not that the two of them ever did anything more than dance on a Thursday. Tomorrow Cissy would join them, but not today. Her friends might send her a message from the bar, but they wouldn’t phone her, not on a Thursday. They knew Thursday was for her parents, and that her parents liked speaking for hours.

One day her girlfriends would know the truth. She hoped they would be happy for her and not angry that she’d lied and kept secrets from them. They, at least, should understand her reasons: the three of them discussed the challenges of an academic career almost as often as they did Petra’s latest man-scapades and probably over the same number of bottles of wine.4 Cissy would be fine, they said, as the most numerate in the Social Science faculty, but they were being nice. Her chances of being invited to do research were slim, of scholarship funding barely infinitesimal. Without a research post she’d have to return to live with her parents in Whitehead, maybe even teach in their school, and she couldn’t face that.

Her friends would understand the reasons but not what she was doing.

From behind the ring-files in her cabinet, she slid out the small photo in its wooden frame. Her secret was Christian Argentstream, and this evening she would have time for him.

In the kitchen she polished the glass cover with a square of paper towel and stood his picture so that she could look at him while she ate. There was an intelligence to his eyes, and something almost heroic to his bearded jaw.

Christian Argentstream (and not Christian Goldbach, the pastor’s son.i) Or to be more precise, the Argentstream Conjecture, deservedly with a capital C. As far as conjectures went, it wasn’t in the top ten, not many mathematicians5 were looking at it and that was a good thing. Not a brand name comparable to a Riemann6 or a Fermat,7 not a hypothesis and definitely not yet a theorem. Not yet. But Cissy had a dream. She’d prove the Conjecture and plant her own mathematical flag in this unconquered plot of prime number landscape. It would make her famous, guarantee her a career in mathematics.8 She had a good feeling about this evening’s Argentstreaming.

The top of her lasagne,9 she realised, was another landscape she had yet to conquer. She dug into it with the knife. Argentstream, I’ll be with you shortly.

The lasagne was oily but it cut easily. When she burned the roof of her mouth on the still bubbling filling, she thought about skipping supper and getting straight to work. But she’d done that before, and hunger had defeated her concentration. She blew onto the lasagne, which made no difference, and then put it into her freezer. She set the egg timer for seven minutes and waited.

Other than her dinner things and Christian Argentstream, the only object on the kitchen table was the rectangular package which she had yet to open. Of re-cut and folded cardboard, once holding Russell Apples, it now had her name and address written in large felt-tip on a white label. The package had been well constructed, its dimensions however uninteresting, almost certainly put together by someone practical but without Cissy’s sense of geometric aesthetic. It could wait a few more minutes.

She leaned toward the package, tempted by a raised corner of packing tape, and was slightly disappointed to detect neither the tartness of apples nor the old book smell of the best libraries, the odour she always associated with wisdom and inspiration. Quiet, respectful libraries, with structured order in stacks, shelves and books, were her favourite destinations and, once there, mathematics texts her favourite books; they often smelled best of all. An old book on Euclid’s geometry became her treasure when she’d taken it home from the library as a child and slept with it under her pillow.

Cissy bent over the package again and sniffed. Nothing. Perhaps they had simply been well packed. An image came of Ellie and Petra at the Blue Island, cocktails in their hands as they watched Cissy sniff a library book. She laughed. They really wouldn’t understand.

She got up. In the bathroom she slid a Tupperware box out from under the towels on the shelf and removed her Euclid, now in a waterproof protective cover, and held it to her nose. Relaxing in her weekly bubble bath the previous Sunday morning, once the water had cooled and the bathroom air had lost some of its dangerous dampness, Cissy had leafed through the book again and reminisced about first seeing its elegant proofs as a nine-year-old. She had become a mathematician on that very day. Maybe just like Einstein had, when he’d been given his copy.

Her thumb had left an oily lasagne-red smudge on the protective cover. She sucked in a breath and yanked a towel off the rack. She wiped the smudge, then the entire book, and replaced it in its plastic box. She looked after her books, better, she was certain, than most libraries. Even among her colleagues in the university library, there were uncaring staff who treated older treasures of wisdom with less respect than the mass-produced first-year texts that could be replaced with a phone call. She’d already had to hold her tongue twice earlier in the week when watching the Deputy Librarian.

Her childhood library had inspired her, but her part-time position in the university library had changed her life. The stacks didn’t have the symmetry of her school library, yet were far more extensive, and still exuded the welcoming, recognisable smells of wisdom in the science section. A few10 weeks in, she had made her discovery.

On the third shelf in the fifth row (both prime11 numbers), she’d found that foxed copy of The Challenges Remainingii with its list of open mathematical questions, problems identified at the turn of the century and still not solved, and she’d been hooked. Most of the topics she’d found incomprehensible – what was the zeta function and why did its zeroes matter? – but Argentstream, now his question she had understood. Prime numbers, everyone knew what those were, even Petra, and you wouldn’t trust her with a bar bill. And his Conjecture, that every large number could be written as the sum of three primes,iii well she’d tested that, mostly by hand, all the way up to a million and three (the first prime over a million). It had been a Tuesday and she’d known she’d cross the million that night, but in the event it had taken her until 02:45 in the morning. The next day was her first time to be late for work.

She’d eaten lasagne that evening also, she remembered, a salty but well-proportioned ready meal. She looked across at her egg timer. It was still whirring contentedly towards its goal.

She couldn’t tell anyone about Argentstream, not until she’d succeeded. She knew what they would think. Especially Petra and Ellie. Popular girls went out; they didn’t do mathematics. Cissy had made a mistake the previous year when she’d told them that she never danced two songs in a row because prime numbers also had gaps between them. The expressions on their faces...she’d recovered by claiming she’d been joking and by dancing without a break for the rest of the evening, but she hadn’t enjoyed herself. She had a better excuse now, tendonitis, and when she didn’t dance she kept an eye on their belongings.

Tendonitis was another harmless white lie, and sitting alone, even with the music thumping, sometimes an Argentstream idea would pop into her head.

In the first months after finding the mathematical challenges magazine, she’d tinkered, addicted to trying combinations of numbers without any real plan.12 Four dozen pads, three gross of pens and two bus stops missed because she’d been adding or subtracting, and she’d not disproved Argentstream nor made any progress. But she didn’t criticise herself for that – it was a bit like dating: you had to kiss a few frogs, hoping for a prince.

Not that the men were much to boast about in this town. Her faculty was almost entirely women, the men owlish or past their sell-bys. The better bowlers at her Singles Ten-Pin couldn’t keep score and the better scorekeepers threw gutter balls and complained of sore arms.

The egg timer buzzed. She rescued the lasagne and the thought of it with frostbite made her giggle. Now the outside was too cold and the inside still as treacherous as integral calculus, but she attacked the food in an organised way, rotating between hot and cold and sometimes sharing both on one fork. She’d learned that from her Argentstreaming: tasks in rotation, in streams, one of which would one day lead her to the silver of discovery. Maybe today, she thought, and took a mouthful, ouch, without first checking the temperature. Concentrate, girl, concentrate.

Argentstreaming was all about concentrating if she wanted any hope for a Eureka moment. Girls’ Nights, her parents and her supervisor’s tasks were the main obstacles,13 plus the monthly Singles Ten-Pin Bowling, although that she went to only bi-monthly, and then just to keep her parents off her back. Last week, marking and a meeting with her supervisor had cost her three evenings – but this week had started well: she’d Argentstreamed twice already and she had a feeling. Even the lasagne tasted different today.

Another bite, more careful this time. She caught herself glancing at the loose corner of tape on the package. Not long now until the evening’s workstream could start.

Argentstream stream #A was testing combinations14 until bedtime, #B was trying a forward-pointing proof, #C the search for a proof by contradiction,iv and #D the sometimes dull but always important searching and reading of literature on the subject.

A proof by contradiction, especially a beautiful one, like the infinity of primes,v now that would be perfect. A proof of some sort was inevitable for the Argentstream Conjecture, just a matter of time until someone completed it, and she wanted to be that someone, to be first to turn the Conjecture into a Theorem,15 and perhaps even have it named after her. It was more than just about her career. She’d be the first Fussgoedel in the history books.

The #A-stream, loyal to mathematics but disloyal to Argentstream, as it could only prove that he had been wrong, didn’t require any subtle mathematical knowledge.16 A discovery would put her on the map but would kill the Conjecture. A result, but not a Theorem bearing her name. A footnote in mathematics history at best.

Today was a #D-day: and the package she’d needed both hands to carry contained two volumes she’d ordered from a library in the capital. Perhaps one would reveal a new technique. She was getting better at finding books and papers, especially with her library job’s access privileges to the national systems. Ungrad at Lee U had taken one of the books out for four weeks, so there must have been something in it, even if the notes on his computer hadn’t yet mentioned his findings.

She finished her lasagne. When she’d washed the dish and cutlery, she stood by the table and cut the strips of tape that held the cardboard packaging together. Inside she found a pristine green paperback and an older grey hardcover.

She carried the paperback to her desk and sat, a pad and pens ready. The first hundred pages brought nothing but a headache and the regret that she did not have anyone who could explain the arguments she was looking at. What had Ungrad seen in it? Only two squiggles on her pad and certainly nothing worthy of the red journal. The volume didn’t even smell good. Her solution, not for the first time, was to record the title on her list of reviewed sources. In the morning she would photocopy the key chapters, and the bibliography, and file it for future attention. Her files were multiplying. Prime numbers might be easy to understand, but progress on proofs was elusive and the theory required to find them intractable and fiendish.

The older book was a long shot. She’d picked it only because of the name Kurt Goedel, perhaps some distant relative from Europe. Goedel had nothing specific to say about Argentstream: it was all general stuff about proofs and proving things. As she read it, once and then a second time, she felt her stomach cramp. To “prove”, same root as the word probe, means to test, yes, she got that, and a counterexample was therefore also a proof, a proof that a conjecture was wrong. That was pretty obvious, OK. So, any conjecture like hers, if untrue, would be “provable” because in theory you could always find a counterexample. Right. That was her Argentstream #A work, looking for such a number.

She was just considering recording this in her notebook when the phone rang. It was too early for a message about a cute guy at the Blue Island. The girls would still be at home selecting their Thursday drinks clothes.

‘Cissy?’

‘Mom?’

‘I wanted to speak to you alone. Without your father hearing.’

‘What’s happened?’ They’d both sounded fine earlier.

‘I received a letter from my library. A complaint. A book you had me borrow came back with pages missing. And they think I did it.’

‘Oh.’ Cissy remembered the book. The photocopier at work had run out of toner and she’d wanted to send the book back before it was overdue. It had only been six pages.

‘Well?’ her mother said. ‘I spoke to them.’

‘Mom.’

‘You did, didn’t you, Cissy? They said the cuts had been very precise. And that’s like you. Like you did with those books when you were at school.’

‘Mom, I’m sorry. I was in a hurry. Our machine was broken.’

‘They want me to pay for a replacement. I’ll send you the bill in the mail and I won’t tell your father this time. At least it wasn’t a book from his school. So he won’t hear about it.’

‘Thanks, Mom.’

‘Are you all right, Cissy? You know I don’t understand your studies but shouldn’t your library have that book anyway if it’s so important?’

‘Sorry.’ No way was she admitting to Mom that she didn’t take math books out of her own library, where someone might see.

‘Sorry? And you haven’t mentioned any boys in our last calls. Is there no one there you like? I bumped into Mrs. Schief last week. She says Brett is doing very well. He’s not with anyone.’

‘Mom. I’m not interested in Brett.’

‘Then you should go out, meet more people.’

‘I’m dancing tomorrow, with Ellie and Petra.’

Her mother was silent for a moment.

‘All right. But be careful. The boys you meet dancing probably aren’t good for you.’

‘I’m not a baby, Mom.’

‘We didn’t put away money for your studies all those years for you to have an accident with one of those... Brett would be better than that.’

‘That’s not happening, Mom. Don’t worry.’

‘And don’t cut up any more books.’

‘Bye, Mom.’

The grey hardback was still open but her mother’s call had distracted her. The section that had made sense only minutes earlier now wouldn’t penetrate. Mathematics was like that sometimes, with comprehension dribbling away like a raindrop on a windowpane. Mom, why did you call just then?

She closed her eyes for a moment, breathed slowly and looked again at the page.

OK, so untrue statements could be proved as such by a counterexample. OK.

Moving on. There were Truths that you could prove, fair enough, but also, Goedel postulated, those that you could not. Whaaat? Her forehead burned and there was a queasy feeling in her stomach. Truths you could never prove? If he was right, then any as yet unproven Conjecture might be in that “true but unprovable” category. Especially a Conjecture that people had been trying to prove for years, like... like... hers. Argentstream’s.

Had she understood that properly?

In the kitchen she poured herself an apple juice but swallowing was uncomfortable, a pressure near her tonsils. She returned to her desk and stared for a while at the grey book lying open to the pages she’d just read.

And by Goedel’s logic, it couldn’t even be determined whether such a Conjecture was provable until, and if, someone actually proved it. So the Argentstream Conjecture might be true but unprovable, and Cissy would never know. She’d been worried that Ungrad or another mathematician would beat her to a proof; she’d kept her secret and invested all that time, just in case it was a race. But the Conjecture might be a race that didn’t even have a finish line, just endless lonely laps.

All those evenings alone dedicated to the future Argentstream-Fussgoedel Theorem.

Evenings when she could have been researching her thesis subject, giving herself a better shot at a scholarship. Or going out with her girlfriends. Or meeting guys that combined brains with good hands. Proving a different conjecture: that there were better men for her than Brett.

Goedel’s result did not mean that there was no proof of the Conjecture.17 She might construct an elegant proof next week. Or that haystack of numbers she was fighting through for a counterexample might conceal a disproving silver needle somewhere, and reveal it on her next calculation.

The haystack was infinite, its bottom unreachable.

She walked to her bathroom, splashed water on her face and rubbed her eyes. She sat on the toilet and waited for her eyes to stop burning. When she stood up, her elbow caught the dangling shower curtain. She looked at its design of tilted wine glasses and red roses, grabbed it in both hands and yanked. A series of plastic pops from the clips and the bundled curtain lay half-in, half-out of the bath.

In the mirror her eyes were red-rimmed and there was a sheen on her forehead.

Maybe tomorrow she would wake up and want to Argentstream, whatever Goedel had said. But not tonight. She rubbed her eyes again and then searched in the back of the bathroom cabinet for the gritty bar of soap her mother had given her. Washing her face felt like applying sandpaper, but she worked the soap all the way to the hairline and even across the sensitive skin under her ears.

She had stopped crying. She carried a chair from the kitchen, climbed onto it and reattached the shower curtain to the pole, clip by clip, each action accompanied by a satisfying snap. In her bedroom she assembled her work and put it into the cabinet.

Christian Argentstream was still upright in his frame on her kitchen table. He too went into the cabinet.

From her closet she selected her most comfortable jeans and the cheerful pink top. In the bathroom her freshly scrubbed face no longer showed any signs of her tears. As she concentrated with her eyebrow pencil, she noticed a warmth in her chest, and she knew that she would not Argentstreamvi again. And she would never have to tell her parents, or Ellie and Petra, how she’d spent so many evenings. No one would ever know. Argentstream was still her secret.

She couldn’t join the two of them at Blue Island tonight. They’d notice something. But Singles Ten-Pin wasn’t due to start for another hour,18 and she could make it if she took a cab.

1The first Fussgoedel Footnote Conjecture argues that the reading of footnotes may provide illumination. A related conjecture is that endnotes are of importance for the more academically inclined reader.

2 Two was a prime number and totalling her lies came to three, another prime.

3 Her supervisor would not approve, but Cissy had calculated that trading marking tasks had rewarded her with an average extra 2.3 valuable hours of free time per week.

4 Cissy had learned that counting wine bottles was unproductive, especially as she was naturally competitive.

5 Ungrad at Lee U was the only one she was sure of, and Cissy knew that he hadn’t done any fresh work for six months. His approach was brutal rather than elegant, which had surprised her given his elfin faculty photo.

6 Riemann is known for a certain Hypothesis (involving the zeta function) ...

7 ... and Fermat for his Last Theorem, previously, of course, a(n) (in)famous Conjecture.

8 If she hadn’t been distracted in her last year of high school by Brett Schief, she would have had the grades for a mathematics degree. Brett was another reason not to return to Whitehead. Brett had large, tanned hands, his fingernails square and well looked after, the way Cissy liked, but he would never understand her. If it weren’t for those hands, he wouldn’t even merit a footnote.

9 Cissy had perfected her own lasagne recipe, but no matter which chef’s knife she used, she couldn’t cut precise rectangles. Asymmetric portions were so unappetising. Industrially prepared lasagne achieved almost perfect edges and today’s portion, measuring 8.1” x 5.0”, was probably as close as any she would have all year to achieving that Fibonacci Golden Ratio in its proportions. A pity that heating it had caused it to lose its shape.

10 It was frustrating that, because of the shift arrangements, she could either count this as the fourth or sixth week. She’d been tempted to average that, to five, but it felt dishonest.

11 A prime number is one that can only be divided by itself and 1. The modern definition excludes 1 itself and therefore the first prime is 2, followed by 3, 5, 7, 11, 13...  Other numbers such as 6 are known as composites, produced by the multiplication of more than one prime (2 x 3).

12 Printing a 10-page list of primes as a resource had occurred to her embarrassingly late.

13 She didn’t count library shifts. After all, without them, there would have been no Argentstream. Nor would she have full Admin privileges on the university networks, and through them to access faculty servers all around the country. Not that she did all of that under her own name. She wasn’t stupid.

14 Many people had spent many years trying many combinations in her #A-stream approach, and so far no one had found an example that showed Argentstream to be wrong. There were days when she thought it was a lottery and that somewhere out there among the billions of combinations, there’d be a winner, one that would disprove Argentstream. But then she’d chastise herself for her emotions. If no one had so far claimed such a winning ticket, wasn’t it actually more likely that there simply was none? And that therefore the Conjecture was true and waiting for her to prove?

15 Andrew Wiles was her idol: he had worked in secret for years, apparently not told a soul, and one day he appeared with a proof of Fermat’s Conjecture. In all the newspapers, the mathematics story of the year, a Conjecture had become a Theorem. If mathematicians had known what he was working on, perhaps a competitor would have been motivated to dedicate more time to it, and have beaten him to a proof.

16 Her #A-stream was mechanical; somewhere out there a computer was probably doing the same thing, and a million times faster, but every number she ticked off her Conjecture list gave her that pop of progress.

17 Two negatives in logical sentences like this do not cancel each other out. It is not multiplication. An unknown unknown is not a known either.

18 This 18th (a composite number: 2x3x3) footnote is dedicated to the second Fussgoedel Footnote Conjecture, which says that there must always be one more footnote than necessary. And to her third, and final conjecture, first published on her Human Statistics blog sixteen years after the night in question, that prime numbers are unlucky. That was the day she read that another mathematician, with the great name of Harald Helfgott, had proven the Conjecture.

i Christian Goldbach is best known for postulating that (rewritten in modern terms) every even natural number greater than two is equal to the sum of two prime numbers. This remains unproven but progress has been made by Ivan Matveyevich Vinogradov, Harald Helfgott and Chen Jing Run.

ii This magazine is fictitious. In 1900 German mathematician David Hilbert published his list of most important unsolved problems, presenting a selection of them at the 1900 International Congress of Mathematicians (ICM). A translated article in English, describing the 23 problems, was published in 1902 in the Bulletin of the American Mathematical Society. The article, however, gave no mention of Argentstream (or Goldbach) or his Conjecture. Accordingly, the Conjecture was at best a second tier unsolved problem. Goldbach’s Conjecture was, however, one of the four unsolved problems of number theory presented by German mathematician Edmund Landau twelve years later at another ICM conference.

iii The standard version of Goldbach’s Conjecture relates to the sum of two primes. Argentstream’s Conjecture, that every large number greater than seven can be written as the sum of three primes, is for all intents and purposes, equivalent to Goldbach’s Weak Conjecture (for which Harald Helfgott advanced a proof in 2013).

iv A proof by contradiction is a backward-pointing proof, where an outcome is assumed and this outcome is then demonstrated to lead to logical consequences that are patently false or absurd. Consequently the assumption must be false.

v How do you prove the proposition that there is an infinity of prime numbers? Assume the opposite: let the highest prime be p. Then think of the number 2x3x5x7x11x13x ...p, the multiplication of all of the primes. Add 1 to create a new number, call it R. Is R divisible by 2 or 3 or 5 or ... all the way to p? No, it cannot be. So either R is a prime number or it is composite and divisible by something else, a number higher than p, another prime. This is a contradiction: therefore, the starting assumption (of the opposite of the proposition) was wrong; and the proposition must be true.

vi For the avoidance of doubt: Christian Argentstream, the mathematician Ungrad and Lee U are fictional. Argentstream was inspired by Christian Goldbach.

About the Author

Carsten ten Brink

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Carsten ten Brink is a writer, artist and photographer. He was born in Germany and raised in Australia, Japan and the United Kingdom. He lives in London and studied Mathematics at the University of Cambridge. He is currently working on a novel set in South America and a collection of short stories. After being awarded a Master of Mathematics, he left academia, but when not writing or making art, he can still sometimes be found, pen in hand, staring at pages of hand-drawn diagrams and scrawled mathematical algebra. He has traveled extensively. He has published both short fiction and photobooks. His photographs have appeared in publications throughout the world.

Read more work by Carsten ten Brink .

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