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Can't Upload Music, So Let's Start with Goldbach

2007/8/12 11:56:07

I was originally going to come online and hold a concert, but the upload kept failing on this website, so let me talk about mathematics instead—let me talk about the Goldbach Conjecture.

For Chinese people, this conjecture is as familiar as can be. But in truth, the Goldbach Conjecture does not hold a particularly important position in mathematics—it cannot remotely compare with the Riemann Hypothesis, the Langlands Program, and the like. However, in entertainment-obsessed China, whoever proves Goldbach would certainly become a huge celebrity, while whoever settles the Langlands Program would probably be known to hardly anyone. That is the difference between amateur and professional. In the professional world, things are always serious, with no entertainment whatsoever.

Take the Langlands Program—forget ordinary people, even ordinary mathematicians basically cannot understand the statement of the problem, let alone solve it. Goldbach, on the other hand, has a statement that even an elementary school student can understand, which is why it gets hyped by so many amateur charlatans. Yet history proves that in the modern mathematical game, which operates at such extreme difficulty, the probability of an amateur having any hope is smaller than the probability of the Earth crashing into the Sun tomorrow. The statement of Fermat's conjecture was simple enough, yet the Englishman's proof—an ordinary person cannot understand even a single line. Even to understand one line would require years of study. Of course, if one could truly understand one line, one could probably understand many lines. But to fully understand the entire thing—there are probably no more than 1,000 people in the world who can.

Regarding Goldbach, the most famous result is Chen's theorem, but essentially, that proof has little to do with Goldbach. Using their methods to prove Goldbach—this ID believes the probability of success is barely higher than the probability of the Earth crashing into the Sun tomorrow. Chen essentially pushed that method to its limit, and over the past few decades, that approach has made essentially no substantive progress. This ID can basically assert: it is a dead end.

Using Chen's method, one can also prove that E(X)—the number of even numbers not exceeding X that cannot be expressed as the sum of two primes—will be far smaller than X^Y, where Y is a positive number less than 1. The upper bound on Y has been continually revised downward; I don't know if there are more recent results. But this method likewise cannot substantively prove the theorem itself. No matter what Y is, it cannot prove that E(X) is finite as X approaches positive infinity. If Goldbach's Conjecture holds, then the number of ways D(n) an even number can be expressed as a sum of primes has an asymptotic formula of T(n)*n/(log n)^2, but this absolutely cannot guarantee that there is no n for which D(n) = 0.

The object that this ID mentioned yesterday regarding the classification of even numbers has nothing to do with specific primes, distributions, sieve methods, or anything of the sort. It is an invariant mathematical structure given by a complex system. As for what exactly it is—of course I cannot say, as it is the core of this ID's method. This object that this ID's method studies, just like groups, rings, fields, and so on, is an abstract mathematical structure newly discovered by this ID, applicable over an infinite range. It just so happens that the prime decomposition of even numbers falls within this structure. Therefore, if the research on this structure can ultimately solve Goldbach, it would be like that young French genius who used abstract structures such as groups and fields to solve the problem of radical solutions for algebraic equations of degree five and above—it would serve as the greatest advertisement for that abstract structure. And the structure itself is far more significant than any specific problem. So this ID's pursuit of Goldbach is not for the sake of Goldbach itself, but for the new abstract structure this ID has discovered.

If this structure's study resolves Goldbach, then just as countless people now study groups, rings, fields, modules, topology, fiber bundles, and so on, in the future countless people would study this new abstract structure discovered by this ID. That is this ID's true intention.

To be honest, this ID somewhat regrets being too excited yesterday—though that is understandable, since years of effort had produced a milestone result. Later, this ID examined the situation from a stock market perspective and realized I am still too lacking in cunning, too guileless by nature. This ID could have done it this way: take those two Diophantine equations, perform equivalent transformations on them, and then offer a bounty—say, 10 million RMB per equation. Then countless people worldwide would work on these two equations, and the final conclusion would probably come much sooner than this ID working alone. Both equations surely have only finitely many solutions—we just do not yet know how to prove it. Once proven, based on the conclusion of that proof, this ID could immediately solve Goldbach. But now, since the matter has already been disclosed, everyone knows that these two equations correspond to something much bigger behind them, so that approach would not work anymore. This ID's interests can no longer be maximized.

Probably many mathematicians would spit blood reading the above, protesting: "How can you approach science with such calculation?" But there is nothing to be done about it—this ID's veins carry not only the blood of mathematics and science but also plenty of the blood of economics and markets, and I have never deigned to conceal this. Either get the whole big apple, or get nothing at all—that is this ID's economic and market principle in scientific research. Just as in the stock market, only this ID drains the market makers' blood, never the reverse. That is the principle. Yes, it is rather stingy and perverse—but that is the principle.