From child prodigy to chief scientist

Xu Xiao wasn't in a hurry to get all the calculation results right away. She took this time to find more relevant papers online and study them.

The calculations were still not finished by evening, and Xu Xiao didn't shut down the computer, deciding to let it run on its own all night.

When Xu Xiao woke up the next day, she finally discovered that all the calculations had been completed.

Forgetting about eating, Xu Xiao immediately started looking at the calculation results.

Just as I had guessed, the calculation results for the prime numbers within these 1000 are mostly less than 2√p.

However, there are a few counterexamples.

Although individual results are only slightly larger than 2√p, such isolated phenomena cannot be ignored as a rigorous conjecture.

"It seems that my assumption was not absolutely true."

It's impossible for Xu Xiao not to be disappointed by this result; it probably means that all of Xu Xiao's efforts over the past few days have been in vain.

However, he quickly rallied, realizing that things weren't as bad as they seemed.

In the process of scientific research, it is perfectly normal for hypotheses to fail.

Even if the assumption is wrong, it doesn't necessarily mean that all the work he did before was in vain.

As long as new and valuable conclusions can be drawn from it, the results of this research project will not be affected.

With his strong mathematical intuition, Xu Xiao made some observations and indeed discovered a new pattern.

"These prime numbers... seem to be able to be further divided into two categories based on the different remainders when divided by 4."

According to Xu Xiao's observations and statistics, for all prime numbers p≡3 mod 4, the calculated T value is not only less than 2√p, but even strictly less than √p.

For prime numbers where p ≡ 1 mod 4, the T value is mostly between √p and 2√p, with only a few slightly greater than 2√p, but still much less than 3√p.

"So, I should actually categorize and prove it based on the different types of prime numbers?"

In this way, Xu Xiao can break down the entire topic and prove the general case bound and the strong bound when p≡3 mod 4 separately.

Thinking of this, Xu Xiao became excited again and finally saw the light at the end of the tunnel in completely solving the problem.

…………

time flies.

Before we knew it, this year's winter vacation has come to an end.

During the winter break, Xu Xiao received a training notice from the national math competition team. He was scheduled to go to Tsinghua University for training in early March.

The entire training camp was divided into two phases. The first phase lasted for 10 days, during which 15 people were selected from 60 members of the national training team to enter the next round.

These 15 students will then participate in a second phase of 9-day intensive training, and only 6 of them will ultimately qualify for the national team to represent China at the IMO.

Actually, Xu Xiao wasn't very interested in participating in the IMO.

His initial motivation for signing up for the math competition was primarily to try and secure a guaranteed admission to university, or at least to get a reduction in the required admission score.

Now that he has achieved his goal, the subsequent training and selection process is not particularly meaningful for Xu Xiao personally.

Of course, as a member of the training team, he will definitely still participate in the training, but there is no need to prepare in advance.

Before the winter break ended, Xu Xiao continued to advance the proof work of the research topic step by step.

For the general case, Xu Xiao's core approach is similar to his previous methods, mainly using Fourier decomposition, Passevar equation optimization and scaling.

Specifically, the proof can be completed by first using Fourier decomposition to split the sum, then using Gaussian sum to simplify the inner sum, and finally using triangle inequalities and Passevar scaling.

The proof for the strong bound when p≡3 mod 4 is much more complicated, employing many advanced algebraic and number theory techniques.

However, this does include some elementary methods, such as using singular features for variable substitution, and using symmetry to cancel out cross terms after squaring.

"Phew... the proof is finally complete. I wonder if there were any problems with the proof process, and how much academic value it has."

After all, compared to the initial assumptions I made, the actual conclusions don't seem as perfect.

Just as Xu Xiao was about to organize the proof process and send it to Yuan Jiyang, she suddenly felt a voice in her mind.

[The host has completed the "flow" task—concentrating on at least 6 hours of focused study or work each day for 30 consecutive days, and independently solving a complex mathematical problem during the task period.]

[Entry obtained: Flow]

[The entry is merging with the host...]

Upon seeing these message notifications, Xu Xiao couldn't help but feel a sense of surprise and delight.

"Is it finished...? It seems my proof process should be fine."

The fact that the system has recognized Xu Xiao’s proof is sufficient to demonstrate the correctness of his proof process. Moreover, the difficulty of this problem also meets the system’s requirements for “complex mathematical problems”.

In his excitement, Xu Xiao temporarily forgot about organizing the results and just wanted to experience the effects of the new entries as soon as the merging was complete.

Feeling that the entries had been integrated sufficiently, Xu Xiao took out the paper he hadn't finished reading before and began to study it carefully.

Strictly speaking, the term "flow" is a level two term. Only by maintaining a focused state first can one potentially enter a higher level of flow.

After maintaining his focus for a while, Xu Xiao quickly sensed a change in his state.

This state was very strange. At this moment, he felt as if he existed in an independent little world, and everything outside became blurred.

The sounds of vehicles driving by outside the window, as well as the chirping of birds, were completely shut out of Xu Xiao's consciousness.

The concept of time is also completely distorted in this state.

Xu Xiao felt like only a dozen minutes had passed, but when she glanced at the clock, she realized that more than an hour had actually gone by.

"So... this is what true flow is?"

Xu Xiao had previously thought that the high level of focus brought about by concentrating on a specific term could already be called a state of flow.

In fact, the effects of the "flow" keyword are significantly greater than those of the "focus" keyword.

This allows Xu Xiao to further improve efficiency by 20% on top of the existing focus on specific keywords.

Of course, at the same time, maintaining the state of flow will further increase the rate at which Xu Xiao's energy is consumed.

Fortunately, Xu Xiao is not lacking in energy. Thanks to the effect of the "full of energy" attribute, even if there is an additional energy consumption, she can recover quickly.