In a World “Primerily” Evil, a short story
[Explanatory note] The following short story explores in a narrative, creative and enjoyable way the mathematical proof that there are infinite prime numbers. The strategy consists in using the proof by contradiction. Let us suppose for the purposes of the proof that there are finite prime numbers.
We could multiply all of them (P1 × P2 × ... × Pn) and obtain a number which would be composite, for it is divisible by all the primes. Nonetheless, if we add 1 to that result (P1 × P2 × ... × Pn + 1) we would obtain a number which is both prime and non-prime at the same time, which is a contradiction. On the one hand, the number obtained is prime because it is not divisible by any prime, for 1 will always be the remainder. On the other hand, however, the number can't be prime, because if it were, we would have multiplied it in the initial operation. Given the incoherence of the initial premise, we conclude that there must be infinite prime numbers. This simple proof is what this story explores by analogy. Using language, I try to personify the marvelous concepts that are numbers, and I breathe life into the abstract lungs of their mathematical reality.
In a World "Primerily" Evil
A long time ago, in a land just like ours, where the sun shines, the wind blows, and the waves break against the evening shore, there was a kingdom. This kingdom, however, was made of peculiar individuals: numbers were people and people were numbers. Their origin remains uncertain until today, but rumor has it that in the beginning there were only ones on the land. The number one was savage and uncivilized, a lonely barbarian who lacked speech and lived in the depths of the woods and on the peaks of the mountains. It is said that civilization arose from some sort of divine intervention and its fusion with the ones... maybe a spirit who descended into the physical world. In short, no one really knows how this kingdom came into being, but what is clear is that it surely existed.
There were different numbers in this land, but in this our kingdom there were only natural numbers. The other rationals and irrationals lived in the far reaches of the empire, beyond the clouds of the nothingness of the negative numbers. The irrational numbers were strange creatures, who did not know what civilization is, and were incomprehensible even to themselves. There was a social order ruling the empire of the natural numbers, which was the fruit of their different natures. The prime numbers were proud of their nobility, for they knew the reality that they were indivisible, a perfect unit. Not only did they live in a perfect harmony in their own individuality, but their harmony was also shared in the society at large, for the prime numbers, precisely, were the ones who governed this empire, they being the nobles, the ones who knew the truth and understood themselves.
The rest of the citizens constituted the other social class and was oppressed by the prime elite. These were the non-prime numbers, the composite. They, unfortunately, did not enjoy such balance, such harmony in their own identity. Their nature was - as the prime numbers used to say - slightly more fallen, slightly less perfect. They ignored their origin and, even though there were some gigantic numbers among them - like 79,309,662 or 35,987,406 -, they still lacked the chivalrous refinement of being prime. And they were well aware of this their miserable reality.
From amongst the noble prime numbers a leader had risen: a king, the King. He was obviously a prime number, but what's more, he was the biggest prime number in the whole kingdom. In fact, people were totally convinced that there was no one greater than him and prime at the same time in the entire world. He bragged on knowing every single prime on earth and thus he defended his right to the throne.
In those times there were still some ones around the lands of the kingdom. These were wild marauders who, being speechless, had not met life in society. They did not even understand what life means. Ones were not self-conscious, just like brute animals. And in fact, the citizens of the empire, both primes and composites, considered them thusly. Nevertheless, there was a one who was different. This one was a miracle of nature. He had been able to learn to talk, and enjoyed a sharp and rational mind, just as the other numbers. He was a beggar and traveled through the empire sowing tears of sorrow. Hoping to inspire pity in the hearts of the other numbers, he used to tell a story: the sad story of his existence. He spoke of how he had to fight for his survival and had been able to overcome his very own fallen nature. One had managed to be more than a one.
He suffered greatly in the hands of the non-prime peasants, who saw in the ones creatures even more insignificant than themselves. The prime numbers also persecuted them, for they used to sneak into their parties and ravage their pantries. Nevertheless, for all the other ones this abuse was nothing more than a mere perception, a simple circumstance in their primitive struggle for survival, but for our One this persecution was a true reality. He was conscious of his own suffering! So miserable was this One that even the gods pitied him.
One was walking dark through the lonely woods on a certain evening, and he stopped and stood towards the west raising his eyes to contemplate the beauty of the sunset sprinkling its rays through the crying branches of the thin trees. And it was then when hope sparked in his life. It was then when this divine being - an angel, a goddess - appeared to him. She was the number Two. The first prime number. It is said that she emerged a very long time ago, at the beginning of times, from the fusion of two savages who once fell in love. Two was the number - the person - who represented love itself: compassion in its greatest meaning. She appeared to our character and showed him a way: to unite himself to her and thus create the number Three. One would die, but he would have offspring. Together, One and Two, would create a prime number who would no longer be as miserable as a poor savage. Their offspring would be a demigod.
Number One, however, in his nothingness, did not want to die. In fact, he hated the prime numbers so much that he would not like his son to become one of them. He wanted to continue living. Two, admiring his desire,granted him the wisdom of knowing how to become something greater without losing his own individuality, without uniting himself to the goddess. This consisted in gathering tears from other numbers and creating a magical potion; according to Two, by drinking it, all those numbers would multiply between each other and they would add to the one who drinks the beverage. Having heard this, the heart of One was filled with joy... and wickedness. Although he had been civilized, this savage being still had a seed of evil in his interior. He conceived a malevolent plan.
One thanked the goddess and went off to all the different towns of the kingdom doing what he knew best: begging. He went to every single prime number in the empire and begged as only he knew how: he told his story, his famous and moving story - his powerful weapon. With this sad story he made the noble primes cry, for even though they were selfish and despicable, they still had passions and enjoyed being touched by such stories. They did not give to poor One any money, not even a piece of bread. The primes only cried while listening to the story, but they did not even console One after learning of his sorrowful reality. They were only looking for an enjoyment in the display of their passions, but then they mercilessly kicked him out. Nevertheless, One was smarter than this. He settled for this treatment as long as he got their tears. One was carrying a small crystal bottle well hidden in between his old and ripped clothes, and there he stored the tears that he collected from the wet cheeks of the noble primes.
Our character did this with every single prime in the kingdom. He even went to the king and begged at his knees, narrating, as usual, his marvelous story. And again, he cried, and One took a tear from his lord, the evil and proud king who despised everyone else... the biggest prime in the world. The most noble person on the face of the earth, in social status obviously, but certainly not in virtue.
Once One had gathered tears from all the prime numbers, he happily set out on a journey beyond the limits of the kingdom, even beyond the limits drawn by geographers and the horizons admired by explorers. He crossed mountains, rivers, deserts, seas and oceans. He pierced clouds on the backs of dragons and other creatures that only children would believe in, until he reached the cabin of the wizard. After traveling in circles for a long time, One arrived at the cottage of the legendary π, a transcendental number, the most powerful sorcerer of the earth. π was the most famous wizard and extremely hard to find. Without a doubt number One had to suffer. But thanks to his nothingness, he had been able to go unnoticed all the way from the royal palace to the confines of the earth. Monsters and other numbers would leave him alone thinking he was another simple savage who had accidentally escaped from his natural habitat.
One also begged the wizard, but this time he was not interested in his tears, just in the magical drink. His story moved the merciful heart of the wizard, who ignored the mischievous plan that the beggar had in mind, and One convinced him to create the potion, just as Two had told him. π magically fused the tears, multiplying in this way all the numbers contained in the bottle, and gave it to One. The potion certainly worked. All the prime numbers of the kingdom multiplied each by the other, creating a gigantic number, yet not prime, for it was divisible by every single prime. Then One shook the beverage and drank it up, adding that enormous number to himself. He laughed and laughed on seeing his great power. He was no longer a one. He was the greatest number on the face of the earth and prime at the same time. One had become the most powerful being that nature had ever seen.
A few months later a great commotion had risen among the royal family and the council of noblemen of the kingdom. They were questioning the king's legitimacy to the throne. Some of the primes were arguing that it was not possible that he could be sure of being the greatest of all. They were convinced that there must be someone or at least some way in which someone could become a prime number higher than their highness. The king felt insulted by this growing attitude and decided to give a party in his palace. He invited every single prime that existed - or at least that he knew existed - even the palindromic and twin primes! Having summoned everyone, he there demonstrated to the people that he was the greatest prime number, the legitimate king. He showed that he knew every prime of the world and that he was sure of everyone else's inferiority.
Suddenly, just as the guests were starting to clap at the king's wonder, the doors of the palace opened with a loud noise. There, in a cape made of bear skin, with a sword hanging from his belt and a fierce look on his eyes, stood he who was formerly One, now a glorious prime, larger than any other in the kingdom. He now called himself Primus. His face was marked by scars and a divine glow shone from his mighty beard. The palace fell silent. Then Primus laughed at the king and shouted in a deep and echoing voice:
"O king, my naïve king! You, in your pride and blind hatred, ignore reality. You do not know the prime numbers. You think you are the greatest but behold, here I am: a prime and greater than anyone."
The king did not understand. He fell on his knees, confused and shocked: "How? Who? When?"
He could not conceive what had just happened, neither could the crowd of primes. They did not know what to do.
"How is that possible? You can't be a prime number. Explain yourself!" shouted one of the primes.
"Indeed I am!" answered Primus. "For I am not divisible by any number, as I shall heretofore prove. For I in my genius and you in your folly, stole the tears from each one of you, and thereby multiplying together you and adding you to me, I have come to be Primus: if you divide me by any prime there will always be one left over. And so, you see? It was that for which you always scoffed at me, my oneness, that in fact is my victory and your defeat."
Finally, the king accepted his failure and exiled himself to a far-off land. Primus was made king and became as despicable as any other prime. In fact, realizing that anyone could repeat his trick, he ordered his armies to kill the wizard π. He understood that, in reality, there are infinite primes, for any savage is a prime in potency. In fact, outside the kingdom most assuredly those primes existed. Still, any one could make a new magical potion as he did, and drink it, thus becoming a greater prime. But if π dies, that would never happen. And just in case, if by some scheming of Two or some other divinity, π would be brought back to life, Primus made a vow to himself never to shed a tear for even the most pitiful of creatures. Primus was no longer that good savage, that benevolent and suffering creature which one day was pitied by a goddess. He was the greatest number in the supposedly finite list of primes. He was king. He was prime. He was evil.