Labyrinthine Prime

My name is Ebenezer, but my friends affectionately call me Ben. Regrettably, I haven’t seen any of my old mates in a very long time, as it has now been 607 days since my incarceration in this godforsaken, labyrinthine, and altogether alien prison. After all that time, my strange jailor’s daily expectations became abundantly clear. If I refused to reveal my vast secret knowledge, I didn’t eat. Fortunately, that had only happened in the inaugural days of my internment, since once I had deduced the pattern, I had been granted access to a per diem, albeit meager meal, only to be redeposited in my sleeping chambers to repeat the seemingly never-ending process the very next morning.

Today began like every other. There wasn’t much of a sunrise in this foreign land to which I had been abducted, but it seemed like it flickered and shifted from utter darkness to glaring light in mere seconds. It wasn’t long before my warden’s sickly blue, rubbery, and protracted tentacles descended upon my doorless, windowless, metal cell from someplace high above. The monster’s limbs wrapped themselves tightly about my body, immobilizing me and lifting me up. I gasped for air in the alien’s grip and another appendage covered my eyes, blocking my vision, as I was transported through the atmosphere of this extraterrestrial world. I had never seen the face of my captor, but I only imagined a horribly massive tooth-filled and vicious maw. I was relieved when the creature released me, and once again I found myself in the center of a sprawling maze; my daily tasks to earn my paltry dinner were about to commence.

The tangled corridors of this labyrinth were far from uniform; some of them were narrower than others, and some of them barely had enough clearance to squeeze through my malnourished frame. However, after months of repetition, I knew my way. The first obstacles I had to navigate were 2 deep, seemingly bottomless, pits. I never desired to discover what might be at the bottom of these crevasses; solid rock, spikes, a deep pool of water? To me it didn’t matter, because I knew they were close enough together that with a running jump I could take exactly 2 additional steps before making another leap.

Once across the gaps, I crawled on my belly like a snake in order to locate and traverse the 3 trapdoors. I hugged my face and nose to the floor, not so much to sniff out the openings, but to feel them. I had long ago discovered that slight drafts could be felt flowing up from the cracks around these hidden doors to tickle the soft skin of my cheeks beneath my lengthening whiskers. At the last gateway, I carefully dropped down to the lowest basement as the secret door flapped shut above me.

Next came a most difficult task; I had to perfectly time moving through a lengthy corridor containing 5 sequential spinning blade traps, with so little space between them that they had to be negotiated as a group. Every set of metallic scythes rotated with a different frequency, and every-other set rotated in the opposite direction to the ones adjacent. My nose twitched, smelling the residual blood of others like me, both on and around the impediments. For several minutes I watched the array of death until the proper moment; in an instant I scooted myself through and beyond.

Checking that I was unscathed, and with a sigh of relief, I mechanically travelled the final winding passageways and intersections of the maze of horror until I came to a door containing a numbered keypad. The keys of this monstrosity were each bigger than one of my hands, arranged sequentially in three rows with a large zero key across the bottom, and every time I lingered here, I remembered the first time I pushed the wrong digit.

That first day, a jolt of electricity coursed through my body until I had pushed the number 7, and I was then extracted from the complex by the same blue tentacles and dumped into my lonely cubical abode without supper. The second time I made it through the maze, I remembered the number that had released me, and so I pushed it again. This time the number 7 also wracked my starving body in pain, and after several more failures, a pattern formed in my mathematical mind.

I realized that the tortuous labyrinth contained 2 pits, 3 trapdoors, 5 sets of fan blades, and the exit first opened on a 7. I thought about it; it couldn’t be 9, as clearly it wasn’t a series of odd numbers because, after all, the sequence started with the number 2. Suddenly the thought occurred to me, it could be a series of prime numbers: 2, 3, 5, 7, and 11! I pushed the 1 and didn’t receive a shock, so I pushed it again to enter the number 11. Lightning stunned me once again, and in frustration I frantically lashed out, pressing 1-2…zap! Then again, 1-3…I heard a click and the locked door opened. Unfortunately my answer had come too late, and once again I was tossed into my cell without nourishment.

That third night I didn’t sleep a wink, as I tried desperately to understand the pattern: 2, 3, 5, 7, and 13? So it wasn’t prime numbers, but what was it? If it was a sequence of primes, 11 would’ve been the answer. Perhaps a sequence of Mersenne primes? No, that would be 3, 7, and 31. Then, just before dawn, the answer bubbled up from my innate and profound understanding of the mathematics of creation. Mersenne primes are prime numbers that can be expressed as (2^p - 1), for some positive integer p. The sequence was p…2, 3, 5, 7, 13, and 17! The next day confirmed my guess, and I received a welcome meal for my efforts. The next several days continued to confirm my solution…19, 31, and 61.

After keying in the answer for day 51…82589933, I knew it didn’t matter which buttons I henceforth pushed. I wouldn’t be electrocuted, because it was clear that my jailor didn’t know the next numbers in the sequence! It was strange to me that a being so advanced, as to be able to abduct me, and hold me in isolation as a slave for so long, didn’t know the simple sequence I was gradually producing for its primitive mind each and every day.

Well, today was day 607, and if my calculations were correct, the p that I would divulge would perhaps win my freedom. Mersenne primes and perfect numbers are deeply intertwined. A perfect number is equal to the sum of its divisors, excluding the number itself. All perfect numbers, much like Mersenne primes, must be able to be expressed by a formula: (2^{p - 1})*(2^p - 1). Until today, all the perfect numbers linked to the Mersenne primes associated with the values for p that I’d graciously provided had resulted in even perfect numbers. Today, however, this one would be the first and only odd perfect number!

My tiny fingers danced across the gargantuan buttons in the sequence I had memorized the previous evening, and after several minutes my solution had been meticulously entered into the alien’s system. As expected, I hadn’t experienced a single shock, but beyond my wildest hopes, when the chalky blue tentacles enveloped me, I was deposited into a cage with several of my long-lost brothers!

With intense tears of joy, I embraced them, and praised my creator for giving me the instinctive math skills to earn my freedom. After all, 607 days is quite a long time when the average lifespan of a white lab rat is at best 3 years!