The God Equation

by Michael A. R. Co

“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.” — Leonhard Euler

We reached the Lionheart Oil just as Matthew Cheng was weighing anchor. The sloop-rigged yacht had mainsail and jib already hoisted. Raffy introduces me to Matthew, who helps me board. He looks like Bruce Lee, only darker and larger. We exchange pleasantries and settle down on the deck. Raffy speeds off.

“I thought you might have forgotten my invitation,” he says, handing me a cold can of beer. A laptop computer rests on the table beside a bowl of potato chips.

“I wouldn’t miss this for the world,” I say, with an American accent. “Boats fascinate me, although I never learned how to sail.”

“It’s a cinch if you have an autopilot,” he says, and we both laugh, knowing full well that it still takes a lot of manual skill to circumnavigate the world solo. Physically, we’re near the same age, but he has the manners of an older gentleman, not condescending, but definitely arrogant. He shows me his laptop. “I customized the software for this rugged little model, so I can monitor every activity on the ship. It not only tracks weather information and GPS coordinates, but the status of every line and tackle through a network of sensors. I can trim the sails from anywhere. It’s my remote control. And best of all–” He toggles to a different screen. “I can connect to the Internet.”

He asks me to wait while he checks the status of an upload and scans a few email messages. Meanwhile, I prepare the mp3 voice recorder that Raffy lent me, and take snapshots with a digital camera.

Matthew turns his computer around and shows me an online chess game in progress. “What do you think the best move is? White to play.”

I examine the virtual board. It was the middle game, both sides showing remarkable symmetry, a clear example of grandmaster play.

“Pawn takes pawn,” I say.

“Excellent call,” he says, and enters the algebraic equivalent. “It’s wonderful that I can always find strong players online. My opponent is from St. Petersburg. Of course, I’m not always sure that I’m playing against a male, female, or human being. I believe chess programs were the first to pass the Turing test. You like chess?”

“I hold my own.”

“Well, we should play someday.”


“So Diego — can I call you Diego? — what shall we talk about?”

“I’m writing an article about the groundbreaking work by theoretical physicists and pure mathematicians from around Asia. I’ve already spoken to some folks in India and China. I was surprised when Raffy told me that there’s actually serious work being done in the Philippines. I hear you’re writing a book.”

“Not yet finished, but I’ve found a publisher. We Filipinos can accomplish miracles if we apply ourselves. Do you have a math background?”

“I’m a science journalist,” I reply. “Stanford.”

“Quick quiz then: how many sides does a circle have?”

“That’s a trick question.”

“Good answer. A circle can have zero, one, two, or infinitely many sides, depending on how you define a ‘side,’ correct? It’s also a shape that exists only in our heads, a perfect shape. The term ‘perfect circle’ is redundant. No object in nature comes close to being a circle, but you see it everywhere, the moon, the sun, all are crude approximations of a concept. But what a concept! All circles, regardless of size, have the same ratio between its circumference and its diameter.”

“Pi,” I say.

“Leonhard Euler popularized the use of the Greek letter, but it was known since ancient times. In fact, for centuries, mathematicians would try to ‘square a circle’ — to create a square from a circle of precisely equal areas with nothing but a ruler and compass — until it was proven that this cannot be done. Because pi, a quantity needed to calculate the area, is a transcendental number. It goes on forever … in a non-random sequence, but with no discernable pattern. More chips?”

“No thanks.”

“A circle is often used as a metaphor for boundaries and enclosures, and indeed, pi was used as shorthand for ‘periphery.’ With all transcendental numbers, one can’t help suspect that there might be a hidden message coded somewhere in the sequence. Like what Carl Sagan suggested in the last chapter of Contact.”

“You think there’s a steganographic message? Perhaps from God?”

“Well, not the kind of message that you’re probably thinking. You can find any sequence of numbers in pi if you look hard enough, even your telephone number. Look here.” He types ‘88888888’ in his laptop, and he shows me that his pi program found the sequence appearing at the 46,663,520th decimal place. “The message is more subtle. In effect it’s saying, ‘There is a pattern somewhere, but you’ll just have to look harder.’ So it’s no surprise that work continues on unlocking the secrets of pi. Funny how the most complex structures can derive from the simplest things. Human beings from single cells, the entire universe from a singularity. Seems a convincing argument against entropy.”

He points to the mainsail. “See that pattern printed on the sail’s edge?”

I notice a colorful, paisley-like design, swirling toward the center.

“It’s called ‘Sea Horse Valley’ and I generated the image using the simple function z equals z-squared plus c.”

“The Mandelbrot set,” I say, pronouncing the name in the German-style, not French.

“A microscopic part of it. I sometimes refer to the Lionheart Oil as the Mandel-boat. The entire fractal image was discovered only in the 70’s after the introduction of computers. Breathtaking, isn’t it? It goes on forever, exhibiting self-symmetry as you increase the magnification. Analogous to the revolutions of the planets or the movement of atomic particles. Check this out.” He bares his left forearm. A well-done approximation of the M-set was tattooed using several inks from elbow to wrist, like a Rorschach blob, the disk and cardioid appearing like the head and thorax of a large insect. “Like it? I did it myself.”

This guy’s hardcore.

Our conversation turned to chaos theory, deterministic systems, Brownian motions, stochastic processes, random walks and whether these walks were truly random. “Randomness is conventionally accepted as true, but no one has been able to prove it. Because if you think about it, it’s actually difficult to consistently produce a random number in the real world. Humans can’t do it because of our psychological predisposition to create patterns, and machines have to use pseudo-random generators for expediency.”

“What about the Heisenberg Uncertainty Principle? Wouldn’t that build a strong case for the existence of randomness?”

“Just because you can’t predict the outcome doesn’t mean it’s random. And what does ‘random’ mean anyway? Godel’s Incompleteness Theorem tells us that you can’t prove certain ideas using the rules within the formal system. You have to go outside, out of the box. Take the case of prime numbers, integers whose only factors are one and itself. Many methods have been developed to obtain or check primes, but it’s still notoriously difficult to factorize numbers. It’s mostly trial and error. We math geeks actually call these ‘hard’ problems for lack of a better word. Indeed the world’s cryptographic infrastructure depends on the difficulty in factoring prime numbers. However, although it’s proven that there are an infinite number of primes, it hasn’t been proven that there is not a general formula somewhere that can factorize any positive integer of any size. When that happens, current encryption systems will be rendered useless.”

He opens another can of beer, and offers me another.

“I’m convinced that there’s a pattern in the sequence of primes,” he says. “Not a message, but something more profound. Unlike the Fibonacci sequence, prime numbers do not appear in nature. They were discovered because Man played with this.” He taps his temple.

He then taps a few keys in his laptop. “Have you heard about the Prime Spirals of Stanislaw Ulam? His findings were first published in your magazine back in the 60’s.”

“Heard but never seen,” I lie.

“If you create a spiral of integers from one to infinity using a square grid, and mark all the prime numbers, you’ll find that they tend to line up along the diagonal axis. It’s amazing.” The LCD screen displays a dark field with white pixels scattered like stars forming delicate lattices of slanted line segments that remind me of city lights when viewed from high above. “I sometimes wonder if this is part of the face of the divine,” he says. “Or maybe they’re just the spots you see when you close your eyes. Euler was completely blind in the last seventeen years of his life.”

“Do you believe in God?” I ask.

“Now that’s a trick question,” he says, chuckling. “I want to, but I don’t, which is why I’ve launched this project.”

It starts to drizzle. “Let’s go below deck,” he says. “I’ve something to show you.”

His cabin is filled with electronic equipment. “There’s an apocryphal story,” he began, “about Euler. During the reign of Catherine the Great, Euler threw a challenge to the atheist philosopher Diderot and claimed he had an equation for the proof of God. Euler said, ‘Sir, a plus b raised to n over n equals x. Therefore, God exists! Reply!’ Naturally, Diderot didn’t and was laughed out of the court. This story is bunk, but when I first heard it, I was intrigued. Could it be possible for God to be reduced to a mathematical equation?”

Matthew sat in front of a large flat screen. He launched a command line window and began to type. “Euler didn’t find the God Equation but he did discover an equation that came poetically close.”

When prompted for his username, he types:


“I’ve configured my system to accept certain special characters. This is Euler’s Identity: e raised to i times pi plus one equals zero. It combines the three basic arithmetic operations and five universal mathematical constants. But that’s just for starters.” He types in his password, and hits Enter.

The screen scrolls up with dizzying, unending lines of horizontal computer code. “Behold!” he says. “The God Equation, version 1.0.”

I peer at the screen from behind his shoulder. He doesn’t bother to look at me as he speaks. I slowly draw my gun from under my vest.

He watches the program scroll continuously. “Paul Erdos, one of my other math idols, second only to Euler, would speak about ‘The Book’ … an imaginary tome written by God, containing the most beautiful and elegant mathematical proofs in the universe. My ‘God Equation’ is actually a computer program designed to seek out these equations. But it’s more than an automatic proof generator. I believe I’ve found an algorithm for the human soul.”

“Artificial intelligence,” I remark. I take one step back but keep my gun low.

“There’s nothing artificial about it,” he says. “It’s virtually organic. It uses the idle system resources of computers across the globe, much like the SETI screensaver.”

“You’ve created a worm.”

“I’ve created an answer. As word of this spreads, more people will download the program and contribute to the effort. Imagine the secrets of the universe revealed, relationships clarified, pi, Mandelbrot, e, the sequence of primes, the Riemann Hypothesis, perhaps even physics equations like the Grand Unification Theory.”

“When did you upload this?”

“About ten minutes before you arrived.” The program continues to scroll up. “It’s not perfect, and I’m working on the next version. There might be some bugs in this system, but because of its complexity, it may take years to find and fix.”

I’ve learned enough.

“Suppose,” I say, “God exists but is displeased with what you’re doing, and so He sends the Angel of Death to stop you.”

Matthew laughs. “Why would the ‘all merciful God’ do that?”

I aim my weapon. He sees my reflection.

“Because faith would have no meaning.”

I squeeze the trigger.

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