# Somewhere over the critical line

Maxamillion Polignac was a number. He was prime, and proud of it. One sleepy Sunday morning, with a cup of tea in hand, he opened the newspaper. The major headline shouted, prime club vandalised: composites blamed! Primes had been subject to prejudice for so long even though primes had founded the society. The strong dislike stemmed from primes being factors of composites. Maxamillion sighed with despair, but resigned, he continued reading.

He aimlessly scanned the newspaper until something caught his eye: the critical line: fact or fiction? The article speculated that the Critical Line was a fabled golden brick road that could lead to the Formula connecting all primes. It was supposedly hidden in the zeta landscape: an untouched land that defined the real and imaginary axes. The zeta landscape was complex and hard to navigate, and hence, the perfect place to hide the Formula. With the Formula, all the secrets underlying primes and how to find them would be revealed. Being a prime, Maxamillion did not have siblings, and living in a conflicted society, he felt alone, despite his many friends. But the Formula would give him a chance to find his twin prime who would make him feel complete.

With this idea on his mind, he went to his friend, another prime named Bernhard Oblong. Maxamillion said “I think we should try and find the Critical Line—we both get something: you’re a factorial prime, hence with the Formula, you could find out what your $n$ is; and I could find my twin prime!”

“Are you sure the Formula exists?” Bernhard asked.

“We have nothing to lose and everything to gain! Just like Pascal’s Wager.” Maxamillion reasoned.

“I hope you are right because I really want to know what $n$ is!” Bernhard agreed enthusiastically.

They embarked on their journey, oblivious to the dangers ahead. Starting in their city, the Number Line, they eventually reached a large building, the RSA bank in the outskirts of town. They noticed a sign with big black letters: no primes allowed.

“Let’s withdraw some money for the journey to the zeta landscape,” suggested Bernhard.

“Sure. But we better be careful,” replied Maxamillion.

A man with a baton stopped them. “Halt!” he said. “Can’t you read? no primes allowed! We don’t have primes coming this side of town.”

Maxamillion and Bernhard had no choice but to leave and try the next village. Another man, wearing a smug smile, saw the commotion through the glass doors of the bank.

“Hmm… primes. Interesting! Primes usually avoid banking with us because we use them to encrypt our credit cards. I know an opportunity when I see one. Let me see if I can capture them,” he thought to himself.

The slippery man phoned the most notorious prime-hunter in the zeta landscape—the Mersennary.

“Hello?” Mersennary crackled.

“I want you to capture two primes headed your way. Bring them back dead or alive. \$1,000,000 in credit cards,” the man ordered. “It’s a deal!” Mersennary replied. Their pockets empty, Maxamillion and Bernhard soon reached a small village. A sign hung over the entrance: \sign{this town is composite-free}. “Wow. Those are some extreme views!” exclaimed Bernhard. Not soon after, they found themselves surrounded by hundreds of primes who Maxamillion recognised immediately: Sophie Germain primes. They resisted any composite prejudice. They were well-built primes$(2p + 1)$rebelling against the system and hoping to teach composites that primes are the building blocks of society, deserving equality. “Who are you? Why are you here? Are you really primes or are you composite sympathisers?” The leader pelted questions faster than the duo could process. “We are Maxamillion and Bernhard. We are trying to find the Critical Line. We are primes and certainly not composite sympathisers.” Maxamillion responded swiftly. “Then you must be admitted into the UPS at once,” the leader proclaimed. “Follow Friedrich into the tent.” The two friends did as they were told. As they were walking, Friedrich explained to them: “The UPS stands for United Prime Service. We are dedicated to protecting the rights of primes against the relentless prejudice of the composites. Join us. We are with the primes, we will continue to be so until the end.” “Consider us members,” Maxamillion said. “Even though you are not as sturdy as us, we could use your help. Find the Formula and put an end to this!” Before she sent them on their quest, the leader gave them a fascinating relic:$\zeta(s)$. “This is the zeta function,” the leader explained. “Use this once you reach the zeta landscape: it will help you navigate your way along the Critical Line.” Progressively, the scenery morphed into a barren land: the zeta landscape. The two dimensions defined the real and imaginary axes. Using the relic, they navigated across the imaginary hills and the complex terrain. Even with hypothetical fog layering the land, they could clearly make out the glowing pathway. There it was—the Critical Line! “Whoa. It really is real. Really real.” Bernhard gasped. However, there was a dilemma ahead, for the Critical Line split into three paths. Suddenly, a prime emerged from the fog. His sunken eyes added to the barren landscape. He whipped out his weapon, the square function. With it, the prime could square Maxamillion and Bernhard and turn them into composites. He advanced towards them, armed and dangerous. The duo trembled as beads of icy sweat trickled down their backs. “I am the Mersennary. I am paid to hunt down primes like you,” he rasped. Maxamillion tried to plead with him: “Why are you trying to break something that can’t be broken? We are all primes here. We have a rich history. Primes have been the dominant species in the whole of maths for hundreds of years. We ruled because we could not be broken down into other numbers. When we multiplied ourselves together, we created composites. Even though the composites have oppressed us, we remain strong and resistant. Primes will never be split. You are one of us, so are you a traitor?” “Sorry. It is nothing personal, just business.” Mersennary responded coldly and inched closer. In a desperate attempt, Maxamillion tried again: “Wait! You are in it for the money, right? War is not a steady business, and I am sure you would earn more at a new job. We want to get the Formula, which could help you learn more about yourself and other primes! You could use that to your financial advantage, eh?” Mersennary pondered and realised he had the wrong end of the number line. He decided to join them in the search. Together they looked at the three new paths:$\operatorname{Li}(x)$,$\pi(x)$, and$x/\!\ln(x)$.$\operatorname{Li}(x)$looked the most promising, because it went the highest, and it looked daunting enough to hide the Formula.$x/\!\ln(x)$was very low, and it seemed like a place to start. “Let’s go$x/\!\ln(x)$!” Mersennary said. “No! Let’s go$\operatorname{Li}(x)$!” Bernhard replied. Maxamillion urged: “Stop arguing! How about we compromise? Let us go explore the stairway$\pi(x)$. Maybe the Formula is hidden in the middle to stop people who aim too high or too low!” They began climbing the never-ending stairs. They were about to give up hope when they saw the Formula. It was$\pi(x)$. When Maxamillion touched it, it surrounded him with a blue light. Full of excitement, he asked the Formula to find his twin prime. But his enthusiasm didn’t last long as the formula would not give an answer. He sighed in desperation, but then he had an idea. He asked it the value of Bernhard. It answered 26951. Then he asked it the value of himself. It answered 26953. “What?! We were twin primes all along?!” Maxamillion shouted. “Wow!” Bernhard exclaimed. He then proclaimed: “With this, we can end prejudice! We could change the composites’ opinion of us by explaining all the secrets behind the primes and how intricate and beautiful we are!” “We could also start a bank that serves all number-kind! Then I would have a steady source of income!” declared Mersennary excitedly. As soon as they got back to Number Line, the trio started a bank: the Riemann bank. Soon it was booming and bought over the RSA bank. The first thing Maxamillion did as CEO was to demolish the sign saying, ‘no primes allowed‘. The law that barred the primes went down with the sign and they both crashed to the ground with a satisfying BANG! The Formula and the relic were placed in Museum Polytechnique. The conflict between the two sets was finally resolved as the composites realised that the Formula revealed the complexity behind the primes. They realised that primes are so complex that they deserve to be treated better. Thus, the numerical landscape was changed forever! #### Glossary  Composites Numbers that can be written as the product of 2 or more primes. Logarithmic integral,$\operatorname{Li}(x)$An approximation of the number of primes until a certain given number, formulated by Gauss. Mersenne primes Primes of the form$2^n – 1$. Factorial prime Primes of the form$n! – 1$. Natural logarithm A logarithm with base$\mathrm{e}$, not base 10. Pascal’s wager A wager that states that if you believe in God and God does not exist, you have nothing to lose. If God does exist, you have everything to gain. Primes Numbers that do not have any factors beside 1 and themselves. Riemann hypothesis Riemann’s conjecture deals with the locations of the solution to Riemann zeta function. It is the holy grail of mathematics. Riemann zeta function An infinite series used to investigate properties of prime numbers. Sophie Germain primes Primes in the form$2p + 1$where$p$is a prime. Twin primes$n$and$(n + 2)\$ are primes.

Tanmay is a 13-year-old living in Seattle, USA. He loves exploring different types of maths (especially ones with a healthy dash of computer science in them). He spends his free time reading comics and playing the acoustic guitar.

• ### A walk on the random side

Tanmay Kulkarni intentionally gets lost on the Tokyo subway
• ### In conversation with Eugenia Cheng

We chat to the author of the best-selling book How to Bake Pi and pioneer of maths on YouTube
• ### Too good to be Truchet

Colin Beveridge looks at different designs for 2- and 3-dimensional tiles
• ### Topological tic-tac-toe

Alex Bolton plays noughts and crosses on unusual surfaces
• ### Mathematics and art: the ELHP

Adam Atkinson uses maths to try to help a sculptor
• ### On the cover: Hydrogen orbitals

Find out more about the weird shapes on the cover of Issue 08