Interest from high up

"The discoveries I made in this last month trumps all my progress of the last decade! It all started by the problem this disciple gave me.

Let n>2 an integer.

Find a, b, c positive integers such that a^n+b^n=c^n.

He told me to prove that there wasn't a solution if I thought so."

Elder Wen took a pause to let the senior have a go at this problem.

The old man closed his eyes and didn't open them for 5 minutes. He started to frown. Suddenly he opened his eyes, that contained a hint of incredulity, awe and even shock. He was much more attentive now.

"Please present your findings junior Wen."

Elder Wen took a deep breath and started to narrate his thought process. "At first I couldn't believe that I couldn't find any solution so I tried and tried. I mean it seems ridiculous that the sum of two cubes can never be a cube, right? And that the same property is also true for all higher powers?

Then I tried to take the problem from another angle. If I have such an integer p and positive numbers a, b, c, such that a^p+b^p=c^p, what special properties would those special numbers have?

I found that you can build this elliptic curve had some bizarre properties:

y^2 = x(x-a^p)(x+b^p)

If I tried to represent this elliptic curve as a rune, it had huge dissymmetries. However, when I tried to project any other elliptic curve to their runic equivalent, they were all very symmetrical. This discovery finally convinced me that this apparently trivial problem couldn't have a solution.

I named those symmetrical runes modular forms, and realized that as a class they have very special properties. Then I proved that modular forms all have an associated unique elliptic curve. Finally, I proved that every elliptic curve has a unique modular form.

So, to come back to the initial problem. If a solution existed, then we could find an elliptic curve that doesn't have a modular form, but we proved that couldn't be possible. So, a solution doesn't exist. I just need to formalize some details and the proof is done.

The implications are quite profound as some rune calculations problems might be translated to their mathematical counterparts to simplify calculations. It makes the visualization of some complex rune interactions easier to approximate and have a general idea of where to search for solutions.

I personally used this method to improve by 50% the coupling of a formation composed of an array altering perception and an array absorbing and reflecting part of a qi attack. This combination had many symmetries to exploit and combine, greatly simplifying those impossible calculations."

The senior who had been listening closely finally opened his mouth. "This trick… is surprisingly brilliant! Who would have thought that traditional mortal math ordinarily only used to train new disciples for rune calculations might find its use in such an intricate way?

This esoteric approach is still rather limited though. Runes are more complicated than just a simple two-dimensional drawing. In most cases this bridge you thought of cannot be used. But it indeed is a great trick for some cases presenting a high degree of symmetry. Rune calculations cannot be translated completely into numbers though. As you indicated, it will remain a tool to restrict the search radius.

Now I want to experiment with your ideas, junior Wen!

Yet, there are still some elements puzzling me. First the original problem. In five minutes, I was able to come to the realization that the problem didn't have a solution by trying to visualize as runes the solution, in a similar approach to your train of thought, I couldn't quite think why that was the case though and your explanation has greatly benefitted me.

However how can your little disciple have the intuition that this problem can't have any solution? If he gave you this problem to solve, it's because he should have already been certain no solution existed. Now I am quite curious about this mysterious genius. Is he even real?"

Elder Wen looked a bit flustered by the sudden question. He didn't dare to lie to this eminent senior though! "He-he is a real disciple but I believe that this disciple has a great mathematical intuition. He might have also found a proof that no solution exists for some values of n. I am sure that this disciple will become an array master in a couple years."

The senior smiled like a benevolent grandfather. "Interesting. Then I will wait for this to meet him. I wouldn't mind having a little discussion with this young fellow when he has learned more about our field."

Elder Wen thought he might try to get some concessions from the senior after his high praise. "May I ask senior, if I could only present the problem and my solution to the assembly and not mention my improvements after I exploited this link?"

The senior nodded. "Sure. That's acceptable, I would perhaps hope for others to try their hand at researching these tools for formations and see if anyone finds different paths to look into.

As for your cultivation, because your contribution to the array profession is not negligeable in my eyes, I will guarantee you that you will be able to reach the peak of core formation in 5 years' time, with a good chance of breaking to the next level in a decade."

Elder Wen was jubilant! This was a dream come true!

He kneeled and placed his forehead on the cold hard ground. "I will be eternally grateful senior."

"Now, now, no need for such formalities between fellow pursuers of the formation's mysteries. And one more thing, take good care of your young disciples. I would be quite disappointed if a tragedy ever befell on a promising talent. Also, you should not advertise his exploits to anyone else." The senior gave a clear warning.

"It will be done senior." Elder Wen responded with gravity.

…

This night, after his traditional cultivating session, Han Lingyun went for a walk under the blanket of darkness. As he left the village of 1st years, soon the sound of nature was his only companion.

Han Lingyun sat down and looked at the clear night sky. This night, Europa wasn't visible and Calypso was only a small croissant. The stars seemed particularly numerous. No light pollution could hinder his front row seat to his favorite spectacle.

Thousands and thousands of stars dotted the darkness, beacons in the immensity of emptiness. Han Lingyun somehow felt hugely refreshed by admiring his inconsequence.

They could be planets like this one on some of them. In fact, he didn't even know much about his own planet. Did it even have a name?

After the non-stop activities of this last month or so, this silent pause came as a much-needed comforting.

He also thought about his parents. In fact, he missed them. In his previous life, he hadn't had the chance to live in such a caring and loving family. Han Lingyun opened his right palm to reveal the bracelet that his mother gave him before he departed.

'Even in the case where I would be identifying myself as 40 years old, I would still miss mother and father. I need to reach qi condensation in under a year to surprise them by coming back home for a visit.'

Han Lingyun completely laid on his back and grabbed the air as if trying to catch some stardust.

'This time, you might not be unattainable. One day I'll really catch you.'

He put his hands behind his head. Be it in Han Lingyun's previous life or this one, he had always enjoyed to observe the world.

The dome of the sky rotated slowly as an hour soon passed by. Han Lingyun definitely didn't recognize any arrangement of stars visible from Earth. However, a brighter strip of stars seemed to divide the night sky.

'This should be a branch of the galaxy we are in. I wonder if I am in the Milky Way, another galaxy or a completely different world.

And what about that Old Man that talked to me before my death. I'm now 99.9% sure that he is not a product of my imagination. He surely came from the stars. He should be a cultivator at a level I cannot fathom yet. If I think about it, granting me some sort of reincarnation, shouldn't I call him a god? I wonder if I will ever cross path with this extraordinary being again. How many are out there in the stars, cultivators with powers beyond my imagination? Ah let's not become too impatient to explore it all.

O but if that old man reached me before, it should mean that there are ways for cultivators to travel through the immensity of space. I wonder why I have never heard about any other planets and cultivators from elsewhere before? In fact, I have yet to hear about anything outside the Sun Empire. Either some higher ups restrict information for some reason, or cultivators here have a too narrow scope.

It doesn't matter, no one will stop me from exploring the universe when I become powerful enough. And now no one can stop me from imagining it to.'

Just as Han Lingyun was having these thoughts, an upside-down face entered his vision, with long brown hair pointing at him curiously from above.

"Eh? Aah!" Han Lingyun rapidly stood up to extricate himself from this vulnerable position of a girl bending down to look at him. He soon came to the realization that he hadn't noticed her approaching at all, he was a bit careless here! Han Lingyun, now on his two feet, tried to determine the features of the intruder. In fact, this wasn't too difficult when his eyes had well adapted to darkness and considering the improvements cultivation brought to his vision.

"Ah, miss Yimo, what are you doing outside at this hour?"

She giggled slightly. "Hmm I should be the one asking you this, disciple Lingyun. My cultivation works best when under the moons. I reckon it's a state of mind sort of thing."

Han Lingyun realized that he had never talked to Wang Yimo, the 5th grade talent, before. He wasn't sure what to say in this situation. "A hmm, you see, hum, I was enjoying the night sky and relaxing a bit. Also, hum, what a peculiar cultivation technique, right? …"

She didn't respond immediately, leaving a blank a bit too long for comfort. Finally, she pointed out. "I didn't know you were the kind to waste time, disciple Lingyun."

Han Lingyun instantly felt that he had to defend himself as though he had just been criticized. "Ahem, I wasn't wasting time, I was working on my dao heart."

Wang Yimo silently examined him, tilting her head slightly to the side.

Han Lingyun was starting to feel more uncomfortable, thank God it was night time! "Well, if that's all, I will be taking a nap. Have a good cultivation session miss Yimo."

Han Lingyun made his way back to his house. None were good talkers and they didn't know each other. Bailing out was Han Lingyun's logical choice to escape all this.

Before entering his house, he took another quick glance at the stars.

'Just wait for me. One day I'll reach you.'