为了科学地解决实际问题，我们必须经常“回过头来”重新研究基本理论，因为只有依靠深刻的理论分析，才能：（1）在表面的混乱中把握规律性；（2）区分本质与非本质现象；（3）预见事变的发展方向。

—— 一位真正的大专家

   用清晰的思想代替盲目的计算。

   Replacing blind calculations by clear ideas.

—— 狄利克雷（Johann Peter Gustav Lejeune Dirichlet）

[笔记，科普，数学] 素数（1）：算术基本定理 fundamental theorem of arithmetic

算术基本定理： fundamental theorem of arithmetic

唯一分解定理： unique factorization theorem

   算术基本定理（Carl Friedrich Gauss, Johann Friedrich Carl Gauss 高斯，1801）

   任何大于 1的正整数，可以唯一的被分解成“一个素数，或多个素数的乘积”。这里，在乘积里，不考虑各个素数的顺序。

   例如：

   6 = 2×3

   20 = 2²×5 = 2×2×5

   42 = 2×3×7

   75 = 3×5² = 3×5×5

   4072 = 2³×509 = 2×2×2×509

   16048 = 2⁴×17×59 = 2×2×2×2×17×59

图1  prime-composite.svg

https://www.mathsisfun.com/numbers/images/prime-composite.svg

图2  Why not continue this list to 100 yourself?

图3  48 的分解 1

https://www.mathsisfun.com/numbers/images/factor-tree-48.svg

图4 48 的分解 2

https://www.mathsisfun.com/numbers/images/factor-tree-48a.svg

图5  Fundamental theorem of arithmetic

https://p16-ehi-sg.gauthstatic.com/tos-alisg-i-6e3a8cj6on-sg/1c051257b132424397a1f119b36b868d~tplv-6e3a8cj6on-10.image

参考资料：

[1] 科普中国，2021-12-31，算术基本定理

https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289929

[2] 科普中国，2021-12-31，惟一分解定理

https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=207079

[3] Vladimir I Arnol'd (Arnold). On teaching mathematics [J]. Russian Mathematical Surveys, 1998, 53(1): 229-234.  Number 1, February 1998

doi:  10.1070/RM1998v053n01ABEH000005

https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005

https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005/pdf

   In exactly the same way a small change in axioms (of which we cannot be completely sure) is capable, generally speaking, of leading to completely different conclusions than those that are obtained from theorems which have been deduced from the accepted axioms. The longer and fancier is the chain of deductions ("proofs"), the less reliable is the final result.

   Complex models are rarely useful (unless for those writing their dissertations).

   与此完全一样的是，公理（那些我们不能完全确定的）的一个小小的改变虽是容许的，一般来说，由那些被接受的公理推出的定理却将导出完全不同的结论。推导的链（即所谓的“证明”）越长越复杂，最后得到的结论可靠性越低。

   复杂的模型几乎毫无用处（除了对那些无聊的专写论文的人）。

   The mathematical technique of modelling consists in ignoring this trouble and speaking about your deductive model as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called "the inconceivable effectiveness of mathematics in the natural sciences" (or "the Wigner principle").

   数学建模方法忽略这些麻烦，把所得到的模型当成真与现实世界相吻合。从自然科学的观点来看, 这种途径是显然不正确的，但却经常导致很多物理上有用的结果，该事实被称为“数学在自然科学中不合理的有效性”（或叫做“维格纳原理”）。    

   Here we can add a remark by I.M. Gel′fand: there exists yet another phenomenon which is comparable in its inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner - this is the equally inconceivable ineffectiveness of mathematics in biology.

   我在此提一下盖尔方德一个观点：还有另一类与魏格纳注意到的数学在物理中不可思议的有效性相仿的现象——即数学在生物学中也同样有不可思议的有效性。

以前的《科学网》相关博文链接：

[1] 2024-11-17 22:51，[数学文化，客观派，讨论] 欧几里得对“素数有无穷多个”研究的有效性

https://blog.sciencenet.cn/blog-107667-1460458.html

[2] 2024-11-10 22:51，[数学文化，笔记] 素数有无穷多个之九类证明

https://blog.sciencenet.cn/blog-107667-1459433.html

[3] 2024-11-02 22:49，[笔记，科普，资料] 素数 prime number 入门

https://blog.sciencenet.cn/blog-107667-1458252.html

[4] 2013-07-23 11:51，孪生素数：相关介绍和链接

https://blog.sciencenet.cn/blog-107667-710546.html

[5] 2024-10-22 22:21，[打听，笔记] 推导符号公式的局限性：从数学、心理学到哲学

https://blog.sciencenet.cn/blog-107667-1456506.html

[6] 2024-05-19 22:49，[羡慕，讨论，物理] 仅推公式就能得到成果的人

https://blog.sciencenet.cn/blog-107667-1434748.html

感谢您的指教！

感谢您指正以上任何错误！

感谢您提供更多的相关资料！