Chapter 16: Gains
Han Lingyun bowed. "Disciple has seen the Elder. This is my major, disciple Lan Ruoxi in 2nd year. She took me on a tour of market road after her advancement to qi condensation. So, I thought we could pay you a visit."
Lan Ruoxi bowed as well to show her respect.
Elder Wen smiled. "You did very well indeed. And congratulations for your progress disciple Ruoxi. Here sit down and taste this Night Leaf Tea."
Elder Wen examined Han Lingyun closely. "Hum, you seem to have been adjusting quite well to the cultivation life during these last few days."
Han Lingyun smiled. "It's okay, but I wish to quickly grow in cultivation to start doing even more interesting things. By the way Elder Wen, when can I start to learn about the field of formations?"
Elder Wen stroked his beard. "Sadly, you would need to be at the qi condensation realm to properly start off. At this stage, you will be able to start manipulating qi to some extent and formulate runes."
Han Lingyun looked a bit dispirited. He suddenly thought of something. "Elder Wen, are these runes the symbols that are described in martial arts when we have to concentrate on a theme?"
Elder Wen smiled. "Indeed, those are runes as well. That makes me remember something. I have some exercises for you to start training in rune visualization. Normally, only those at qi condensation realm could work on this but I'm sure that you could boast some gains at the body strengthening realm. It never hurts to try at least. Here take this book and try to follow the instructions. But don't insist too much if you are not making any progress."
Han Lingyun wasn't expecting too actually profit this way. "Thank you Elder once again. I'll be sure to try the field of formations later on!"
Elder Wen took a sip of tea. "Hehe that's all I wanted to hear. By the way are you still interested is some math questions?"
Han Lingyun nodded with fervor.
Elder Wen smiled. "Okay let's make this more interesting. I'll give you 50 Ps if you can solve it under 15 minutes."
Han Lingyun nearly spit off his tea. "Can you do that?"
Elder Wen patted his shoulder. "Of course, I can! I'm an Elder. As long as I don't show extreme favoritism you are good to go. The only issue is that you don't have a sect token yet so I won't be able to transfer Ps. Oh well, if you win this little bet, I'll just transfer them to you when you receive your badge at the end of the month." This was also a way for Elder Wen to rope Han Lingyun into the array pavilion.
"Okay here, take some paper, a plume and some ink. Listen well:
What is the highest integer n such that (n^3+100)/(n+10) is also an integer?"
Han Lingyun concentrated. First, he tried with some small values of n to get a feel for the expression. Soon he realized that he had to transform the fraction into something easier. He had to fumble around a bit with the expression before finding the trick and showing that the answer was n=890.
Elder Wen clapped. "Haha well done! Well done! You win 50 Ps. Why don't you give me a math problem now for a change?"
Meanwhile, Lan Ruoxi was looking at the whole scene playing out with a blank face. Not only did she feel quite lost but she couldn't believe how easily her little minor just made 50 Ps in 10 minutes. It must be said that a 1st year disciple will gain 1P for every day he comes to class, and that represents the basic income for lowest ranked disciples. Obviously, there are other ways to gain Ps like completing missions, or having a high ranking…
However, to seemingly effortlessly make in 15 minutes what the poorer disciples make in 1 months, and without even having a badge yet??
Under Han Lingyun's guidance, disciple Lan Ruoxi understood another thing today: array masters are rich, and milking them is a direct ticket to the moon.
"Elder Wen, what type of problem do you want?" Han Lingyun really enjoyed this game.
"Hum let's see, how to make it interesting again? What's the hardest problem you can imagine with a simple looking formulation?"
Han Lingyun started to have some very evil thoughts. With his last life's knowledge, Elder Wen had no idea into what he was going into.
He tried to act as casual as possible. "That's sounds like a good idea. That way, it becomes a little competition between the one giving the problem and the one solving it. But obviously, it wouldn't be suitable for me to give my 50 Ps back to an illustrious master like you Elder Wen. Maybe we could be betting on the time you take to find the solution? For example, if you take less than an hour, it's your win and you could ask something from me, but every day you don't find the answer, you give me 50 Ps."
"Not a bad idea little one. However, days? I think you are grossly underestimating the processing power of a core formation cultivator AND an array master." Elder Wen felt like Han Lingyun was looking down on him too much.
"I have a good imagination too." Han Lingyun defended himself.
"Well, we will do as you say. But you better find an interesting problem then, or I would be thoroughly disappointed if I can find the answer in minutes." Elder Wen raised an eyebrow.
"Great! What if, Elder Wen, by any circumstances, i-if somehow, I find a troubling problem and you get stuck on it as the days start to pass by?" Han Lingyun looked a bit sheepish.
Elder Wen acted slightly offended. "Are you doubting the integrity of an Elder of the White Tiger Sect? I have already agreed on your proposition. Not to mention, 50Ps is nothing for me, I can gain thousands in a day. And if you really succeed as you seem to believe, I would still be glad to reward you for it."
Both were truly confident about this little game that raised the stakes.
After a moment of thought, Elder Wen still found Han Lingyun's behavior to be a bit fishy so he decided to take some precautions. "I'll just add two points, you only have 1hour to come up with the problem, and disciple Lan Ruoxi here and I will have to judge if it verifies the conditions we set. Having a simple formulation. Disciple Lan Ruoxi, what did you understand of the little problem I gave to your minor?"
Lan Ruoxi didn't expect to suddenly be pulled into all this. "Hum. I understood the question and I know what basic operations like n^3 mean it's multiplying n with n and again with n. But I'm not quite sure I understand how Han Lingyun came up with the answer."
Elder Wen was satisfied, under these conditions, he was supremely confident. "That's perfect for our purpose."
Han Lingyun acted as though this was a bit troubling. In fact, since he proposed all this, he already had a couple of very promising problems that plagued mathematicians for decades in his past life. Some looked deceptively easy.
"That restricts a lot the type of problem I could imagine. How about doing 50Ps per 12hours to compensate?"
Elder Wen smiled at his antics. "Deal."
Han Lingyun was rejoicing in his head. 'Hehe, I love the sound of that. I still need to make a choice between those problems though. Hmm, the Riemann hypothesis might not fit the conditions of the agreement… The Collatz conjecture is definitely a strong contender, simple and still unsolved, but I would like to give it a shot when my cultivation is higher… Oh yes, I have a good one! This could be a way to judge the importance of cultivation on brainpower as well.'
For the next 30 minutes, Han Lingyun acted as though he calculated different possibilities and was searching. In fact, the only thing he was calculating was how he should use his huge passive earnings of 100 Ps per day!
Finally, he wrote down on a new page.
Let n>2 an integer.
Find a, b, c positive integers such that a^n + b^n = c^n .
"That is my problem Elder, what do you think. Also, if you believe that there are no solutions, you have to prove it."
Elder Wen nodded. "That's a given." They looked at Lan Ruoxi.
Lan Ruoxi examined the paper and furrowed her brows. "I can understand the problem. But isn't that too simple, minor Lingyun? You might want to search for something harder, it's a shame to miss out on such an opportunity.
"I'm satisfied with the problem." Han Lingyun reiterated.
Elder Wen seemed eager to try it out.
Han Lingyun stood up. "Thank you elder for the tea, I should come back to the 1st years housing area for supper and my evening cultivation session."
"Hehe I'm glad you enjoyed it. I'm going to work on your little problem then."
…
As Han Lingyun left the array pavilion with Lan Ruoxi following him, he just couldn't hide his grin. There are no solutions for the problem and proving it is incredibly difficult.
To give some context, in his last life, this problem was first raised by Pierre de Fermat in 1637. Fermat showed that there are no solutions for n=4. During the next two centuries it was also proven for other values of n (like 3, 5,7 ). It took 358 years to show it for every n. The man who succeeded, Andrew Wiles in 1994, isolated himself for 7 years to create this proof. In the process, he actually had to build a whole other branch of mathematics! The proof itself? It's more than 100 pages long. Not to mention, mathematics were vastly more advanced on Earth.
Skipping with joy, Han Lingyung hugged Lan Ruoxi who was bewildered by his actions. "Thank you miss Lan! How about it, I could give you a small comission of 10 Ps a day for being the judge. There wouldn't have been this result without your help!"
Lan Ruoxi shook her head. "I-I have done nothing. I don't deserve anything. It wouldn't sit well for me. But are you really that confident?"
Han Lingyun smirked "Hehe I really am. In fact, I would also be ecstatic if elder Wen ever succeeds. I would pay a ton of Ps to read his findings and proof. But, the more likely scenario, is that I'M RICH BABY!"
CREATORS' THOUGHTS 
This_one_
This_one_ Additional resources for the curious ones:
-Detailed solution for the first problem
https://www.youtube.com/watch?v=3ECQZm9XoAg
-Andrew Wiles proof
http://www.scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf
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