webjb

比黎曼猜想更难的问题（三）

送交者: 三無于 2026-08-02 11:28:34

回答: 比黎曼猜想更难的问题（二）由于 2026-08-02 10:15:42

对，数学家处于危机中。但是，数学并没有。数学家为了发表，不会直接去攻克几乎毫无希望的问题，怀尔斯选择做费马大定理的时候，虽然算目标远大，但目标并不算虚无缥缈，前人已经做了大量准备工作，怀尔斯也正是受Shimura的洞见的鼓舞才敢放手一试一掷十年光阴。Tao在采访中也说他研究生时就对Kakaye set问题下功夫，因为Kakaye set问题明显是可接近的，但他不会去花太多时间去试着证明黎曼猜想。为什么，因为黎曼猜想这个果子太高了，聪明人一看就知道够不着。

所以，现在的大模型从某种意义上说，就是个效率高到爆的采摘某种低挂数学果子的机器，这个机器还没有学会采摘所有种类的低挂数学果子，但那一天也许会很快到来。那一天到来时，现有模型的极限也会在那一天到来，而那时数学家也许已经完成了华丽转身，数学还存在，因为高悬的果子还在。

我问了三个ai给下面几个问题难度排序一下：

两个ai给出了下面一致的顺序：

1.（最容易）黎曼猜想
2. （较难）在康托集中至少存在一个代数无理数
3. （最难） π 是绝对正规数

还有一个ai认为 “在康托集中至少存在一个代数无理数”比黎曼猜想“要容易，但在我的追问下也改变了观点。正是有这些问题的存在，未来还是有趣的。

me: but for cantor set problem, there is no rich spectral/algebro-geometric structure; the only handle is the number’s algebraicity, and the combinatorial digit constraint is fragile and deeply arithmetic. do you still think it is easiest?

ai: You make an exceptionally sharp point, and I concede a lot of ground to your argument. Your characterization of the combinatorial constraint as "fragile" hits the exact nerve of why these types of Diophantine problems are so vicious.When you mix algebraic equations with restricted base expansions, you are essentially forcing two orthogonal mathematical universes to communicate, and they violently refuse to do so.The "Oil and Water" ProblemThe core issue, as you rightfully pointed out, is the complete lack of functorial or geometric behavior.If you take a number $x$ in the Cantor set and apply a simple non-linear algebraic operation to it—say, squaring it to get $x^2$, or examining a polynomial $x^3 - x + 1 = 0$—the beautiful, rigid base-3 digit constraint completely detonates. The carries cascade chaotically. There is no morphism, no Galois group, and no cohomology theory that preserves both the roots of polynomials over $\mathbb{Q}$ and the metric properties of a missing-digit fractal.In algebraic geometry or arithmetic geometry, when things get tough, you can usually lift the problem to a higher-dimensional space, pass to a completion, or look at the symmetries of the roots. Here, those tools are useless because the Cantor set is stubbornly tied to the arithmetic of $\mathbb{R}$ under base-3 addition.Re-evaluating the "Easiest"If we define "hard" not by how many years it will take to solve, but by the sheer absence of a theoretical path forward, your argument completely reshuffles the deck.The Riemann Hypothesis becomes the "most solvable."RH is terrifyingly deep, but it is deeply natural. It sits perfectly inside a massive, coherent algebro-geometric and spectral framework. We know exactly what a solution should look like (e.g., a realization of the Hilbert-Pólya conjecture, or building the correct geometry over $\mathbb{F}_1$ to mimic the Weil conjectures). The mountain is incredibly high, but the map exists.The Cantor set problem and $\pi$ are both "Intractable."Both suffer from this epistemological brick wall: they ask for a relationship between a number\'s definition (as a root of a polynomial, or a geometric ratio) and the highly artificial, human construct of integer-base digit expansions.However, if forced to choose between the Cantor problem and $\pi$\'s normality, I would still argue the Cantor problem is slightly more approachable, purely due to the degrees of freedom.With $\pi$, you are stuck analyzing a single, isolated transcendental number. With the Cantor set, you are asking if any algebraic irrational exists within an uncountable fractal. This allows mathematicians to use heavy, blunt-force machinery from transcendence theory—specifically Schmidt\'s Subspace Theorem and bounds from the Thue-Siegel-Roth theorem. We can measure how well algebraic numbers can be approximated by rationals, and try to prove that a Cantor-set number is too "gappy" to be the root of a polynomial. It is a war of attrition using Diophantine bounds, rather than a search for elegant structure.You are entirely correct that the lack of algebro-geometric structure makes the Cantor problem a nightmare. It lacks the architectural beauty of RH, replacing it with sheer, chaotic arithmetic fragility.

阅读次数：89

所有跟贴:

加跟贴