CHRISTIAN GOLDBACH AND HIS CONJECTURES
By Andrew Paul Grell
Lots of people think they are one in a million or have been told they are one in a million. Caleb Hoffman knew he was one in a million. It was sophomore year, advanced calc. The course started with twenty-one students. After session five, tensor analysis, only seven were left. Caleb estimated the number of students matriculated in advanced math courses at three million, and he multiplied that by one third, the number of surviving classmates. He sure knew, statistically at least, that he was one in a million.
Caleb did not disappoint. Over a Foosball game with a biology major friend, he accepted a commission to analyze peculiarities in genetic changes to leaf structures. Adapting the famous work of G. H. Hardy, a mathematician who had likely never seen the inside of a lab, Caleb wound up doing his doctoral dissertation based on that project and, of course, titling it Laurel and Hardy’s law.
Academic and professional success followed. Caleb took on the long-unsolved Bellhop’s Tip problem. Three cross-country cyclists decide to spring for a hotel room one night. The room was $25. In the morning, after the continental breakfast, they squared up at the desk. Each cyclist put in a ten-dollar bill, and the clerk handed back five singles. They decided they would each get a dollar back and they would tip the Bellman two dollars. They each shelled out $9 and the Bellhop’s tip was $2. Nine times three is twenty-seven, plus a $2 tip is $29. The started with $30, what happened to the missing dollar?
It was the professional success that landed Caleb in the spot he was in now. Although Lobachevsky was certainly “the greatest mathematician to get chalk on his coat,” it was the strong hand and outstretched arm of Caleb using Powerpoint that finally convinced state legislatures to cut the state share of lottery revenues from the current 60% to the casino-normal 7%. A privatization deal funded by Caleb’s Zeta Beta Tau brothers from Wharton bought him the time and space to pursue his dream: proving (or disproving—either way was good) that all even numbers greater than four could be expressed as the sum of two odd prime numbers. Goldbach started kicking this around with Leibniz in 1742, but nobody has proven it. We’ve tested every number up to 400 gazillion without finding a counter-example. Caleb got himself a high-tech, automated yacht and set off alone to cruise around the Pacific while attempting to find a proof. The boat, of course, was red and blue. He was tempted to name his new ride the Grace Hopper for the floating computer power it had, but, since finding the proof would probably take a miracle, he decided on Teva. Ark. Caleb was damned if he was going to christen anything, so he rounded up some of his father’s surviving Coast Guard buddies—the Jewish Navy—and they put together a reasonable facsimile of the ceremony for naming a baby girl. And he threw in an advance prayer of thanksgiving for surviving a sea voyage. Mistake.
Caleb headed north to the Aleutians; as a child, he had always wanted to see for himself the passage between Little Diomode Island in Alaska and Big Diomode Island in Russia. And, being north, he would spend less time sunbathing on deck and more time working. Great in math but not so great in other areas, Caleb found out that being able to sail to the islands meant he couldn’t witness people walking between Russia and the United States, a winter-only deal. He docked long enough for a meal of broiled sea otter and a few hands of Inupiaq poker, supposedly the longest running game in arctic history. He was told that people haven’t walked across since the start of the cold war, but maybe they were bluffing.
It was downhill from there. Caleb teased apart every result proving pieces of the conjecture but could make no further headway. Every island seemed to be the same as every other island, and so he told the autopilot to sail where there were no islands. Next, he tried the Andrew Wiles method—going far afield for theorems in completely unrelated disciplines of math which might be usable for his proof. Wiles proved Fermat’s Last Theorem using knot theory, of all things. But there were now so many fields, finding a good theorem could take forever. Laughing by himself was not a good sign. Caleb remembered back to AP Calc in sophomore year at Bronx Science. If integration by parts doesn’t work, try integration by divine intervention. In what he told himself was an exercise to keep his math chops toned, he programmed the yacht’s computer to calculate the times of the Hebrew morning, afternoon, and evening prayers. He hadn’t touched a prayer book since his Bar Mitzvah, but when he got a cellphone signal, he downloaded a copy. And made a donation of $18.
Now Caleb had a routine. Get up, shower, have breakfast, recite the prayers, go through Algebraic Geometry. Lunch, Email, go through Stochastic Processes, inspect the boat, say the afternoon and evening prayers. Eat dinner, shuffle through other disciplines to pick the next two. Netflix, sleep. No work on Saturday. Repeat. After six weeks of this with no results, he did a last search for disciplines he might not have heard of. Not likely, but there was a non-zero probability that there might be one. That’s when he discovered Bottomology. Timon of Sicily, uncle of Timon of Phlius, model for Shakespeare’s Timon of Athens, started the field after he lost a child who fell off a roof. The field started with the regular solids and conic sections being used as predictors of when and under what circumstances would things fall. And there in the middle of the list of Bottomology theorems were three that involved solid constructions in ratios of prime numbers. Gott in Himel.
The next Sunday, a new moon according to the computer, Caleb recited the Psalms of Thanksgiving and got to work. But not before wondering why the ancient Greeks got to be mathematicians while the ancient Jews were numerologists, with their Gematria. And why name your numerology after geometry? Wouldn’t something like Qalqula be better? It was three more days until Caleb was able to put one of the theorems to work; there was a light at the end of the tunnel. Caleb should have paid more attention to KPMG commercials: the light was an oncoming train. Ten minutes after he knew where he was going, the Teva beached on a rocky shore. Hull stove in. Mast and antenna snapped off. Caleb made it out of the water with his skin and his teeth.
Castaway Log, December 6th, 2019. Crashed into the Island, which I do not believe was in the yacht computer’s database, three days ago. Surveyed Island. I got lucky; it’s a pandanus habitat; the fruit is much easier to deal with than coconuts. Plenty of birds with plenty of eggs. Now I know why Deuteronomy says not to take the mommy bird if you take eggs from a nest. I may be here for a while and I wouldn’t want to run out of eggs. Plenty of fish; fortunately, the emergency kit was salvageable and it included fishing stuff worthy of Popeil himself. What’s left of the boat provides shelter, there are MREs if I get tired of omelets and drupes from the pandanus. No snakes, no large predators. There’s a stream of sorts, which raises the question: am I on a desert island or a deserted island? I never had that clear. I’ve set up a big SOS on the beach from the widest rocks I could carry; I understand from television that’s the thing that’s done. Pandanus leaves soaked in diesel at night, just like Exodus, a pillar fire. My best course of action is really to continue the work on Goldbach rather than waste paper on this log. I’ll make another entry when something important happens.
Castaway log, March 17th, 2020. Dressed in green (leaves) today. Something important happened; I believe I have proven the Goldbach Conjecture, thanks to some help from Timon of Sicily. Too bad I’m cut off from the peer review process. Good thing I didn’t waste paper; the proof is a hundred and twenty pages long. I’m going to spend the next week going through it for flaws.
Castaway Log, March 24th, 2020. Two things happened. Acting as my own jury, I accepted my proof. That’s the good news. The bad news is that when I got down to the beach for a celebratory swim, the SOS rocks had been moved around to spell out “Don’t Publish.” Geckos, as far as I can tell, can neither spell nor lift large rocks. I rearranged the rocks to spell out “Why not?”
Castaway Log, March 25th, 2020. This is becoming a habit. The answer came back “Danger.” I spelled out “To whom?” Now I’m wondering if I had wasted the effort of adding that final “m.” I found the solar panels in the emergency kit and charged up my phone. I set the camera to motion detect mode and tied it to tree branch.
Castaway Log, March 26th, 2020. “To everyone” was spelled out. I looked at the camera app. I was the one spelling out the messages. I spelled out “Why would a number theory proof be dangerous?” I knew the answer, of course. San Bernardino. Sayed Farook. No one could crack the pass code of the terrorist’s iPhone. Because of number theory. But I wanted to see if my subconscious knew that. I also wanted to see if it was really me answering me back or someone who looked like me. Had to rule it out, of course.
Castaway Log, April 1st, 2020. I or my doppelganger had the same answer to why Goldbach was dangerous. I waited a few days and wrote “To go boldly where no one has gone before.” Last night I tied myself to a tree overlooking the beach, put a dangerous dent in my coffee supply, and waited up to see what was what. The knot I used, and if I ever get off this island, desert or deserted or whatever, I would tell the story to Wiles over pizza and beer, was Gordian. I had left one hand marginally free and held a knife in it. I could cut myself out if need be but could not untie the knot. If dawn broke to find the rope intact and the camera showed me moving the rocks around, that me wasn’t me. It didn’t come to that. I saw someone about my height and weight walk onto the beach and start spelling something out. From where I was tied up it looked like “Meet me at Farpoint in two days at noon.”
Caleb used the last bit of soap to shower and shave, and he donned the faux commodore’s uniform his dad’s buddies gave him at the naming ceremony. Caleb had been to Farpoint several times before. It had Daytona Beach sand; if you drew something in it, it would last at least a little while. Caleb had used it as scratch paper. Once more he traversed the jungle backbone of the island and made it to the spit of scrub sticking out due north. The man already waiting there looked like someone from the Seinfeld episode where they were having coffee at the diner and Jerry asked if two people at another table were from the future. Never shy, Caleb marched up to his apparent cohabitant and offered his hand.
“Please mind the drawings in the sand. Caleb Hoffman, mathematician.” The buff, shining blond accepted the proffered hand.
“Frum. John Frum. Caretaker. Guardian.” Caleb considered the name. The most Christian first name was coupled with the Yiddish word for “observant”. Something about the name was tickling his memory.
“And what is that you guard and take care of? Obviously not small craft at sea.”
Frum smiled with the whitest and most even teeth Caleb had ever seen not in a movie.
“There’s a lot of high-level, but low-volume, information spread out between Irkutsk and Tierra Del Fuego. It would be disastrous if it were lost. That’s what I guard. You and your information wound up in my patrol area, so you’re my problem. I don’t seem to be doing a good job. Naked eye stellar navigation is harder than what you do, and I do fairly well at making sure it doesn’t disappear. Got it into the papers a few years back, that gave it a jolt. I was ham-handed with you but you still managed the proof. All you have to do now is cross 122 degrees west and you’re someone else’s problem. Maybe they’ll be too busy to take you on. Maybe they don’t think you’re a problem.
Realization was just starting to dawn; that was pretty slow for Caleb’s rarified cerebellum.
“You’re, um, not from around here, are you? Frum. John Frum. That’s the name of the Cargo Cult deity. I think my dad told me the story, the natives would watch big, noisy birds drop off supplies to white men who never seemed to work or hunt. So they built landing strips with bamboo control towers and waited for their own supplies from the birds, They worshipped John Frum. Dad said the natives didn’t understand the idea of last names, so all the white people were John from Chicago or John from Detroit or whatever.”
“That’s why I took the name. Very good. That’s why this zone needs guarding. These people know when something is going on. East of 122, nobody cares. We have four cable networks showing archeological finds of models of spaceships and astronauts. These are all viewed by people who went to High School, at least. Draw a line from Bombay to Stockholm. Outside of 600 miles from that line everything was Animist until recently. Animist to the west, philosophical to the east. Inside 600 miles, it’s all human-like deities. People from the sky. No one figured this out?”
Caleb, not having a terribly high EQ, was trying to determine if Frum was screwing with him.
“So, John. Where are you from?”
“Pick your favorite Star Trek episode. Gary Seven. Q Continuum. Who Mourns for Adonis. Probably not that last one. You know how difficult it was to get those stories into production?”
Honesty sometimes works wonders.
“I’m trying to tell if you’re screwing with me or not. Someone else could be close to solving Goldbach. Maybe they sent you to sink my boat, and when the other person gets published, I’ll be miraculously rescued. Or not. I don’t even know if I need to know who you are and where you come from. Is there something you want from me, besides not publishing Goldbach? Is there a path that ends with me having a tongue and corned beef sandwich at the Lido Kosher Deli?
Frum looked like he was making a decision. Caleb took that as a good sign; Frum could easily have killed him in the storm. Maybe his crew is opposed to killing people.
“Sophia takes care of the Balkans to Finland. She pegged Kurt Godel when he was 14. Let him peg her as well, so to speak. They remained friends, corresponding on technical issues. It was Sophia who nudged Kurt toward the Incompleteness Theorem. Some things are true which can never be proven to be true. You probably know more about it than I do. She did it so that people could come to the conclusion that Goldbach was one of those true things that can’t be proven and move on fractallettes or something.
Caleb’s face fell. Godel, valiant conqueror of the Lying Cretans and the monopolistic village barbers, had help from his, what? Alien girlfriend? Hyperdimensional Muse?
“You’re still not telling me why Goldbach can’t come out.”
“Goldbach is the key. It’s a lot harder to solve than you think, even though you actually did it. It was as if you picked a perfect 2018 NCAA bracket. Goldbach is the touchstone. Solving Fermat might have been a Master’s thesis. Four-Color Map could have been a homework assignment.” Frum seemed to be getting wound up about this, stopped for a breath, and continued.
“You know how many hucksters there are here, don’t you? People selling infinite this, eternal that, perfection of the other thing? Promises and scams are getting more complicated and easier to pull off. And it’s not just your little rock, here. The only way to know if a promise-maker of the infinite is real is if they can show the proof of Goldbach. I didn’t make that up, your own astrobiologists did. Would you like to hear my offer?”
Wow. negotiating with a messenger angel? A not-so-ancient astronaut? Extraterrestrial con-artist? Hallucination?
“Shoot.” Caleb hoped that wouldn’t be taken literally.
“We’ll give you the solution to Birch and Sinnerton-Dyer. A French freighter will pass by the island tomorrow and see your SOS. They’re almost deadheading, they’ll be happy to hoist your yacht and drop it at their next port of call, San Diego, for a salvage fee from your marine insurance. They might have it put back together by then. You’ll get a Field’s Medal.”
“Ellipses? I could have been drawing them all this time. Well. Okay, I guess it’s time for me to get home.”
= = =
Le Fave and the restored Teva made it to New York at the same time. As promised, Caleb stood the entire crew to a meal of his heritage in Long Beach. And then decided to pick up the tab for the other diners. And handed Wally his Fields Medal check for $15,000.
“Think you can take a Canadian check, Wally?”
In the men’s room before leaving the deli, Caleb looked in the mirror. And saw John Frum.