The twenty-fourth Fermat number is composite. This has been proven by a friend of mine, Ernst Mayer.
The thirty-third Fermat number has 2,585,827,973 digits. It is the smallest Fermat number still in the categorization "unknown" when it comes to prime status. We know it exists. Numbers bigger than it exist. Because life is too precious to write out two and a half billion digits, we can write it shorthand like this:
2233 + 1
We know F33 is out there. F33 is much larger than a googol. A googol is 10100, which is also out there.
Or it could be in here? Would F33 exist if there was nobody around to think about it? My grandmother lived 86 years on this earth and never once did she pause to consider whether F22 was composite, even though someone worked hard to figure out it was.
I doubt I would have been able to describe a prime number to my grandmother, a woman who had only a formal a 4th grade education. But she was able to teach me the Italian word for "teeth", and the song they sung at the start of class every day in the one-room schoolhouse in Messina, Sicily.
Good morning to you
Good morning to you
We're all in our places
With bright happy faces
And this is the way
We start the new day!
In Sicilian school before the second world war it was important to be bright and happy. That was the precursor to learning, the receptiveness to experience that enabled the person to progress through life.
It's hard for us normal people to experience Fermat numbers, and much easier for us to imagine sitting in a hot one-room schoolhouse being taught the alphabet, or riding a mountain bike in the Alps.
Unlike the biking and learning, numbers are like art. One can stand in front of a Joao Miro painting and think about F33 and realize he feels the same way.
What is happening there that makes such things exist? What must people think who sense such unworldly items? I have never experienced F33 or F21 or even F5 which Fermat was sure was prime but turned out not to be. In fact, Fermat thought all the numbers 22n were prime, and he was wrong, but nobody at the Juneau Safeway seems to mind the way they don't mind that Fermat never bothered trying to prove that An + Bn = Cn only works when n=2.
Things like this are worried about by the likes of Ernst Mayer.
Ernst Mayer is not like you. He's an American, born in Germany. Speaks English with no trace of an accent, and I imagine his German is the same, as is, I suppose, his French, Italian, Japanese, and Norwegian.
He recently sent out a missive to his spam list making light of a situation with a player on the Ghanaian soccer team whose name is Kaka. Turns out Mr. Kaka is a rather excellent player, so when he played Spain, the Spanish sportscaster had to call out his name frequently.
Kaka is the colloquial Spanish term for feces.
Ernst, ever the wordster and having just watched the Spain vs. Ghana game (in Spanish, another language Ernst speaks) felt it was hilarious hearing the Spanish sportscaster stutter as he shouted, "GOOOOOOOOOALLLLLL de KAKA!"
And he's also discovered very large prime numbers.
Ernst is a quadriplegic due to a spinal column injury he suffered as a young man. He was in a mountain biking accident which he remembers vividly. Going down a grassy hill into a gully in the Alps he misjudged the steepness of the uphill that faced him. He endo'ed (went over the handlebars), landed on his head, broke his back and injured his spinal column.
Ernst gets around in a wheelchair better than most people do on feet. He seems to be propelled by a lightness of being that cannot be imposed upon by the physics of the world. Ernst would be bright and cheerful and brilliant if he was crushed under a falling locomotive and only two Ernst molecules were intact.
I had the distinct pleasure of working with Ernst for two years in my first startup company. Then I went away to Antarctica several times, and worked for a different company. Recently, I ran into Ernst at the sushi place right across the street from Apple Computer HQ. We shook hands and vowed to call.
When Ernst caught me up with his life he mentioned he'd just got out of the hospital. Naturally, I figured he'd undergone some experimental nerve repair surgery, but alas, this was not the case.
Ernst was crossing DeAnza Boulevard when he was hit by a car, driven by a person who chose not to stop for the red light. Not stopping for red lights is such a problem in Cupertino, California, that traffic cameras have been installed at many intersections and the fines for being caught are extremely stiff. Getting people not to run red lights in Cupertino is apparently as difficult as convincing a mako shark not to attack a bloody wounded seal.
The car that plowed into Ernst did not kill him, but it destroyed his electric wheelchair and broke almost every bone in his body, including the ones under which he still had muscular control.
When I spoke to Ernst about the accident he was himself. He harbored no ill will for the driver. He was not angry at life for dealing him such punishing blows.
Instead, he put me back on his spam mail list which he uses to distribute his puns and diatribes against authority figures who do not follow the rudimentary rules of human decency and compassion.
I have in front of me on my desk a copy of the paper, "The Twenty-Fourth Fermat Number is Composite," published in the "Mathematics of Computation", volume 72, number 243, December 6, 2002. I got it from Ernst's website, which encourages you to download a screensaver that will put a demon on your machine that will aid in the computation of Mersenne primes. In typical Ernst humor, he points out that he's competing with SETI at home, but to date, even though the ramifications of SETI are much larger, the possibility for success of his prime search is much greater. After all, we know there are ever larger prime numbers out there, while we may never hear "peep" from our galactic neighbors.
I downloaded the paper because I visited his website, because I got this message from Ernst a few days ago and it reminded me of a time when I thought my entire life was going upward and to the right, when I wasn't expecting to be dealt anything heavy, or that I'd ever worry about money again.
Recently I felt my life was crumbling to ashes around me.
I wasn't hit by a car while crossing the street in my electric wheelchair.
Ernst Mayer: Goooooooooooooooooooooooooolllllllllllllllllllllllllllllllllllllllllllllll ... golazo! golazo! golazo! Un golazo espectular de ... es correcto? Eres seguro? - de Kaka! El gran gol de Kaka en el 89. minutooooo..."
It's "Kakaesque", I tell you.
Life is good.
My paternal grandfather haled from a town in Sicily called Florida, pronounced - FLOR-REE-DAH. He was a barber for most of his life. He painted pictures. He was prolific. He sold many paintings to his barbershop customers. Many he gave to his family.
My brother and I grew up in a home surrounded by my grandfather's art. There were at least three of his pictures in our bedroom.
My grandfather painted two things: the Italian countryside and clowns.
He painted his landscapes from memory. When he painted a picture of a broken down farmhouse that might have contained a mill, he'd paint the word, "MILL" on the outside, so there'd be no doubt in the mind of the viewer. When he painted his clowns, they were characters from Puccini opera.
When my brother grew up he became a professional artist. One day, in passing, we got to talking about our grandfather's art.
"It's not very good," my brother said. "He's got no sense of perspective."
"It looks okay to me," I said, and he gave me a squinty-eyed sneer.
"And the color. Ever notice every one of his paintings uses the same colors?"
I shook my head. Again, the sneer. I said, "But he's self taught. Not so bad for a guy who never took a lesson in his life."
"He definitely has talent. But it never went anywhere."
"He was a barber," I said.
"He's a barber."
A few years ago my brother got some money and took my grandfather back to the town of Florida, Sicily. When the town fathers found out my grandfather was returning from the new world, they planned a big celebration. My brother had a bunch of my grandfather's paintings shipped over, and the town put them on exhibition in city hall. They put banners on the streetlamps and draped a big one in front of city hall the way museums do when they're announcing the arrival of the King Tut artifacts.
People came for miles to see the pictures of clowns and the Italian countryside and to shake hands with my old grandfather.
At the reception my grandfather glowed, and the town residents fawned over his paintings. My brother was there, enjoying the ambience. Many people asked about the locations in the pictures, and my grandfather assured them they were all local to Florida even though none of the people recognized them.
"They loved his stuff," my brother told me when he got back.
"But you said it's not any good."
"And they know it. They're all art lovers out there. They know good art when they see it, and they know bad art. It wasn't about the art. It was about grandpa coming back after 75 years in America."
"The whole town pretended the clown pictures were good just to welcome him home after all those years?"
"He was just a barber who painted beside the cash register when he had no customers."