# Ikshvaku and the 163-adic Udupi Shri Krishna Bhavan

*Achyut Bharadwaj*

##### December 2023

This is a short story that I initially wrote for my UChicago supplemental essay for the following prompt:

*“If there’s a limited amount of matter in the universe, how can Olive Garden (along with other restaurants and their concepts of food infinity) offer truly unlimited soup, salad, and breadsticks? Explain this using any method of analysis you wish—physics, biology, economics, history, theology… the options, as you can tell, are endless.” – Inspired by Yoonseo Lee, Class of 2023.*

I then transmorgified it to a $2$-adic metric space and shortened it to make it look more like an essay rather than a story before submitting, although I like the story form better. So here goes!

Having trekked in the cold and snowy mountains of Ladakh without any food for a day, Ikshvaku is desperately looking for a restaurant. Born and brought up in Bengaluru, where the weather is always pleasant and restaurants abundant, Ikshvaku realizes that restaurants are unlikely to exist in a desert five thousand meters above sea level in temperatures of -20 degrees celsius. Yet, he wants a Masala Dosa, a traditional south Indian breakfast item. After many frustrated hours of trying to get google maps to work, he accidentally stumbles upon a hut. It has a signboard labeled “$163$-adic Udupi Shri Krishna Bhavan.” Udupi Shri Krishna Bhavan is a chain of restaurants that serves traditional South Indian food. Used to seeing Krishna Bhavans in the unlikeliest of places, Ikshvaku does not find this surprising at all.

He pushes through the door and enters. It is surprisingly warm. It feels like a different world inside the outwardly small hut. He notices a live music band entertaining customers with their music. He finds it odd. He had never before come across an Udupi restaurant with live music let alone with a rock band playing electric guitars and drums. Listening to the music keenly and closely, he notices that the singers sound more like distorted guitars with their voices fading in and out, and the lyrics strangely out of order. He thinks he knows the carnatic music they’re playing, but the notes that reach him feel like they’re jumping forward and backward in time.

The tables in the restaurant also seem weirdly distributed, clumped together in rings, almost on top of one-another in some places, and separated by wide open spaces that apparently serve no purpose. The hut is surprisingly large and appears almost endless. It’s almost as if it extends to infinity, and yet, it feels like a closed space. A point far far away on the horizon seemed to circle back and end close behind him! He’s eager to wander around and explore more, but first, the Masala Dosa! He settles down at a table and quickly zeros in on the Masala Dosa. Incredibly, he finds “Masala Dosa Infinity” on the menu! As many refills of dosa as you want, for the cost of two dosas! Figuring that the owner must have got his statistical simulations wrong, and taking upon himself the challenge of depriving the restaurant of all its Masala Dosa, he orders the “Masala Dosa Infinity”. He’s too hungry to be bothered by the fact that the waiter he beckoned from the far end of the hut magically appeared by his side much faster than it would have taken anyone or anything he had ever known to cover that distance.

Wolfing down his first serving, Ikshvaku begins thinking about how the restaurant is going to keep its side of the bargain if he were to continue ordering dosas till the end of time. Surely they’d run out of batter? Suddenly, he hears the voice of his high school math teacher talking about sequences and series and sums to infinity. Of course! It all falls in place in a trice. The restaurant, he reasons, is going to halve the quantity of their next serving, and continue to do that for every serving! So $1+\frac{1}{2}+\frac{1}{4}+…$ to infinity is just going to add up to $2$ – exactly what they were going to bill him for! So that’s the trick.

Excited at uncovering their devious plot and slightly disappointed that he couldn’t fault them for it, he beckons to the waiter again for his second serving. Surprisingly, his second dosa is one hundred sixty three times the size of his first, and his third dosa is twenty six thousand five hundred sixty nine times its size!

“How in the world do you manage this?” he asks the waiter.

“Sir,” says the waiter smiling, “this is a $163$-adic restaurant. Our restaurant prides itself on keeping its word!”

“$163$-adic? What does that even mean?” he asks. With a sigh, the waiter begins to explain.

“You know how in your world you have the real numbers? And you define the distance between them as the absolute value of the difference between them?” asked the waiter. There’s a hint of disdain in the waiter’s voice when he voices “your world.”

“Of course,” says Ikshvaku, ignoring the tone. “That’s elementary.”

“Well, we do it a little differently in our world – the world you entered when you stepped into our little hut. The way we define distance is completely different – we use the inverse of the maximum power of $163$ that divides a number to define distance. So let’s say we’re in your world and we can measure things only in one dimension along a line. Let’s say I’m at point “$3$” and you’re at point “$12992244$.” In your world, the “distance” between us would be the absolute difference, that is $|12992244-3| = 12992241$ units.

“Of course,” says Ikshvaku again. “That’s obviously the natural way to do things.”

“But in our world,” says the waiter, “the $12992241$ units can be written as $163^3\times3$, and $163$ divides the number a maximum of $3$ times. So the distance is $1/163^3$. Now, if you were at point $705911764$ instead of $12992244$, then $|705911764-3| = 705911761$ units, which is $163^4\times1$, and $163$ divides it a maximum of $4$ times, which gives a distance of $1/163^4$ (which is closer than the previous distance of $1/163^3$ for $12992241$).

“You’re telling me that something that’s $705911761$ units away is closer to me than something that’s $12992241$ units away? You must be crazy!” says Ikshvaku.

“We think about distances a little differently,” says the waiter. “Let me show you:

$705911761$ in $p$-adic base-$163$ | $010000$ |

$0$ in $p$-adic base-$163$ | $000000$ |

$12992241$ in $p$-adic base-$163$ | $011000$ |

“Which is closer to zero? Going from right-to-left along the number, the $705911761$ is more like zero and all the lower digits correlate until we reach the fifth digit. The $12992241$ on the other hand, quickly diverges away from zero from the fourth digit. So in our world, we regard $705911761$ closer to zero than $12992241$. We can use any base, $p$ where $p$ is a prime, and they become the $p$-adic number system. We settled on $163$ because the number system $\mathbb Z[\sqrt{-163}]$ is a unique factorization domain."

“But why would you ever want to do something like that?” asks Ikshvaku, flabbergasted.

“Oh, we’re a bunch of mathematicians and physicists who built this world. We triggered a measurement collapse at a different point of probability on the wave equation that was a superposition of the $p$-adic and Euclidean universes, and forked ourselves off from your universe. We rebuilt it on more elegant principles that can handle inconveniences such as negative numbers, imaginary numbers, and infinity. That’s how we serve you dosas that keep increasing in size and still converge elegantly without violating any physical laws. Up there by the fireplace are Cauchy and Schrodinger, if you’d like to have a lively discussion on the $p$-adic system and parallel universes.

As he struggled to process the concepts of measurement collapses and convergence, Ikshvaku’s mind was awhirl. So in this world, $1+2+4+8+\cdots$ converged to a finite $p$-adic number. Suddenly everything he’d experienced seemed to make better sense. The weird clumping of everything around him in rings – he was inside a $163$-adic metric space, where $1, 3, 5, 7, 9, \dots$ were all the same (maximal) distance away from him, while $163, 326, 489, 652, \dots$ were one sixty third the distance and each of these constant-distance spaces appeared like rings. A point in space very far away was like a small negative distance, so space appeared to circle back behind him! The denizens of this universe with the neural nets in their brain natively interpreting the $p$-adics, must perceive it as natural, while he, with his biological systems from a non-$p$-adic universe must be seeing these jumps and discontinuities in space and time. He watched in fascination as the waiter turned away and suddenly teleported to the edge of the hut before jumping back and forth between the different “rings” at one-sixty-third, four million three hundred thirdy thousand seven hundred forty seventh, and seven hundred five million nine hundred eleven thousand seven hundred sixty one-th distances away.

His mind still awhirl, Ikshvaku continued downing his 163x-magnifying dosas. He soon lost count of the servings but strangely didn’t feel very full. In fact, he actually felt lighter. Deciding finally that he’d had enough, Ikshvaku beckons the waiter again for the bill. “Of course,” says the waiter, and brings him a tray with a neatly arranged stack of strange-looking currency notes. “What’s this?” asks Ikshvaku. I think I owe money, not the other way around.”

The waiter smiles, and shows him the bill. On the bill is a single statement: $$1+163+163^2+163^3+163^4+163^5+\cdots=-1/162$$

Legend has it that Ikshvaku, simultaneously confused and delighted by the new metric space, never found the door of the portal back to the “real” universe, and spent the rest of his life eating dosas at the $163$-adic Udupi Krishna Bhavan. He attained immortality, experiencing forever the superposition of the $p$-adic and the real, confusion and delight, and of being hungry and full.