Mersenne’s Mistakeby Jason Earls |
Part 1 appears in this issue. |
| conclusion |
“My sorcery tells me many things. Yesterday you wrote a letter to Frenicle de Bessy concerning prime numbers. Your letter contained many mistakes. In your sequence of values for primes of the form 2p-1, you made the claim that 267-1 and 2257-1 are prime. But they are composite. You set forth a foolish hypothesis that your superficial human knowledge has no way of ascertaining.”
“Perhaps I did. But your criticism does not bother me.”
“Surely you don’t think mathematics is exempt from my sorcery? Allow me to demonstrate my expertise.”
Mersenne’s eyes widened. He wondered what the sorcerer would say next.
The tiny man bent toward the creature in his arms. He leaned down and whispered into its ear. It released a soft whine and raised three twisted arms to the sorcerer’s face. Mersenne counted eight fingers on each hand.
“I will construct a prime from your mistakes, Mersenne. Keep in mind your non-prime values of 267-1 and 2257-1.” The sorcerer shut his eyes. He placed his fingertips to his eyelids and began to sway. Mersenne didn’t blink.
“If you concatenate the values of 267-1 and 2257-1 with 68 zeros between them, you will get a prime with the decimal expansion of
14757395258967641292700000000000000000000000000
0000000000000000000000000000000000000000002315841784746323908471419700173758
15706539969331281128078915168015826259279871.”
The sorcerer recited each digit.
Mersenne wiped sweat from his chin. “Are you positive that is accurate? Is that really a prime number?”
“Of course. I can give you more if you like.”
“Yes! I mean...” Mersenne tried to force away his curiosity. He lowered his head.
The sorcerer shut his eyes and floated in a wider arc. He concentrated so hard he rose to the center of Mersenne’s cell. The creature in his arms squirmed and moved its mouth awkwardly. Mersenne noticed thin insect-like wings folded under its back.
“I see the pattern in the numbers now,” said the sorcerer. “It is complicated and buried deep within their natural order, but I can see the correct values that make them prime.
“Let 267-1 be r and 2257-1 be s. The values of n that make r * 10n+s prime are 42, 62, 146, 210, 936, and 1490, with no others up to 4000. It means that with values greater than 62 — (because the first two values don’t yield the desired decimal expansions) — if you concatenate the r and s values while inserting 68, 132, 858, and 1412 zeros between them, it produces prime numbers.”
Mersenne stepped backward and shook his head. He went to his desk and dipped his quill and wrote down the sorcerer’s values with a note of explanation beneath.
He turned and faced the floating man and suddenly felt anger burn through his body. “Why are you here? What do you want from me?”
“I want my magic to flourish, of course. I want a pestilence to spread throughout the land. I want to dispose of all Christians. I want demons and magicians and Satan himself to freely roam the Earth.”
“Never. I will do everything within my power to prevent that from happening.”
The creature moved its many appendages and the sorcerer laughed. “You actually think you possess the power to stop me?” His eyes brightened.
“Yes. I do.”
The magician stared at him. He looked beyond Mersenne’s eyes and into his mind and saw that Marin did have sufficient power to hinder his efforts.
“Unlimited mathematical knowledge,” the sorcerer said.
“What?”
“I will give you unlimited mathematical knowledge.”
He glanced to his writing desk. “I would never accept anything from you.”
“And in return you will end your anti-witchcraft studies.”
“Nothing you could ever give me would stop me from trying to annihilate you and your kind.”
The sorcerer raised one hand, extended a finger signaling Mersenne to wait. He bent toward the deformed creature and whispered into its ear again and Mersenne saw that both of their eyes were glowing green.
The sorcerer chanted a phrase in an unknown language and Mersenne’s arms began to shake. A burning sensation passed through his body. His legs felt heavy; he couldn’t move. A dark presence travelled up his spinal column and he felt a change occur inside him.
“You now have the gift,” said the sorcerer.
“What gift? I told you I wanted nothing from you.”
“You don’t have a choice. It is done. You will notice the change the next time you write.”
“No, you can’t force any change upon me. I am a man of God.”
“It doesn’t matter. Evil is stronger than good.”
“Never. You are wrong.”
“It doesn’t matter. You have the gift now. You must cease your anti-witchcraft studies. We will leave your land and practice in another part of the world, and you will never try to stop us again.”
Mersenne started to respond, but knew arguing further was futile. He turned his back to the sorcerer, dropped to one knee and began to pray. In the background he could hear the creature choking and gasping.
The sorcerer whispered: “What shall we do now?”
“R-repeat the d-digits of the f-first prime to b-bring him d-down,” the creature said.
Mersenne stayed in his kneeling position and listened to the sorcerer recite every digit of the initial prime’s decimal expansion again. He tried to ignore the digits and continue praying, but his appeal to God wasn’t strong enough to prevent the sorcerer’s magic from working. The sounds of the digits forced his hands to drop, his knees gave way, and he collapsed to the floor.
The sorcerer floated to Mersenne’s writing table where he rifled through the manuscripts. He took the Bible and letters and tucked them under his arm. He recited more words in the ancient language, then floated out the window with the creature.
* * *
The next morning Mersenne lay on his wooden bed. He couldn’t remember what had happened. He stared at the ceiling, feeling the hard wood beneath him. Images of the sorcerer and the creature slowly returned. He tried to determine if the encounter had been a dream or not.
He leaned up, glanced around his cell, noticed his Bible and letters missing. Why would they want them? He stood and retied his robe, then went to the dining hall for breakfast, never mentioning the sorcerer or the creature to his fellow monks.
An hour later, back in his cell, he took up his quill and began a new letter. “Dear Frenicle, last night I had a strange dream. I don’t fully understand what happened, but I think a sorcerer and his...”
His hand started spasming.
The quill dropped.
His fingers cramped and froze into painful positions. His hand lurched for the pen, grasped it, and furiously scribbled out the formulae for several large numbers. Gargantuan integers, the numbers consisting of at least a million digits. He felt no control over his hand and it wrote extravagant claims beneath each formula. “This is the first odd perfect number. Difficult to find because it has exactly 463,918 digits...”
Other formulae and mathematical claims followed, more astonishing than the previous. His hand listed sophisticated algorithms for factoring large integers and more definitions for strange numbers followed with full descriptions of their properties. He did not recognize or understand all the claims. A few things he remembered hearing from Frenicle de Bessy a few years ago, which at the time he did not fully comprehend.
Mersenne forgot about his letter to Fermat. His hand scribbled on and his mind absorbed the information. Numerical patterns and algebraic conundrums flowed and his body seemed detached from his hand’s actions.
Then he realized the encounter with the sorcerer had not been a dream. His hand stopped writing and he stared at the formulae and algorithms and claims before him. They slowly began to make sense. He could see their structure. See the logic underlying each one.
The abyss of pure number theory was revealing itself to him with perfect clarity, and he became frightened, then exhilarated. The cold logic, the harrowing certainty, the absolute truth of pure mathematical form. The profound structure humans were not supposed to observe.
He shook his head, tossed down the quill and stood up. His hand lurched for it. He grabbed the hand with his other one and stepped toward the window. I will cut it off, he thought. I will slice it off at the wrist.
Then his body moved back to the desk and a force pulled him down into the chair. He took up the quill and put it to the parchment and wrote, “Formula for Primes,” then a long string of symbols and instructions beneath.
He immediately recognized that the formula would produce each prime in consecutive order.
The formula he had spent his life searching for.
He investigated the formula and it only required fifteen minutes to determine its accuracy. He managed to write out the first 20 primes with exactly 1000 digits each.
After four hours of mathematical work, Mersenne’s hand stopped writing. He regained control and began another letter to Fermat. He wanted to write of the mathematical claims and theorems he had seen, but mainly he wanted to relate the formula for primes the sorcerer had given him. To tell him how simple and elegant it was.
But he set down the quill after a few minutes, realizing he wouldn’t be able to reveal the knowledge to anyone. He couldn’t let them know since the gift came from an evil source. He would lose their respect. They would realize the knowledge was the devil’s work.
He pondered how he might be able to reveal the prime formula with subtle hints. Perhaps I can work it out from a different angle, he thought. But after two hours of trying, he realized it was impossible.
Finally he lay down his quill and went to the window of his cell. Images of formulae and numerical patterns lingered in his mind. He looked out at the fields for awhile, then leaned his head against the glass, shut his eyes and prayed.
When he looked up again, he saw the creature the sorcerer had held during the encounter, hovering ten feet from his window. its thin insect-like wings fluttering and its mouth opening and closing in grotesque shapes. It still had no eyes. The creature raised its three arms with eight fingers, holding them out and reaching for Mersenne. But Marin covered his face and turned away. A few minutes later he looked up. The creature was gone.
* * *
Mersenne stopped writing to his scientific correspondents after receiving the sorcerer’s gift. He spent all his time investigating the formula for primes and other mathematical enigmas. Each time he would pick up his quill, his hand would produce yet another deep mathematical insight or theorem that would arouse his curiosity and compel him to investigate. Many of his former correspondents wrote to enquire about his sudden end to their discussions, but he never responded, even though he wanted to share his new knowledge with them.
Marin Mersenne was the only man to have ever known the formula for producing consecutive prime numbers.
But in 1648, it died with him.
Copyright © 2008 by Jason Earls
[Author’s note] Jason Earls’ most recent novel, Cocoon of Terror, is now available from Afterbirth Books.
