数学 1801

《算术研究》

卡尔·弗里德里希·高斯

他把算术变成一面钟，奠定了现代数论。

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In depth · the introduction

卡尔·弗里德里希·高斯拿起整数这个杂乱的世界，发现它其实像一面钟一样运转。

核心思想

在钟面上，9 + 5 不是 14——而是 2。过了 12，就绕回起点。高斯把这种「绕回」变得既精确又普适。任取一个数 n，叫它「模数」，此后两个数若除以 n 留下相同的余数，就算「一样」。他写作 a ≡ b (mod n)，并把 n 称为模数——这个记号，乃至「模数」这个词，我们今天仍在用。

听起来不起眼，却改变了一切。一个关于无穷多个整数的问题，化成了一个只有 n 个刻度的有限钟面上的问题，而原本看不见的规律忽然就显形了。高斯整本书，都从这一招搭起：在钟上做算术。

它是怎么来的

高斯出身于不伦瑞克一个贫寒之家，很早就被发现天赋，并得到当地公爵的资助。十八岁那年，他证明了两千年来无人做到的事：仅用直尺和圆规，就能画出正十七边形——这件事让他最终选择了数学，而非语言学。他私下记着一本发现日记，二十一岁时便写成了这部书。

在他之前，数论是一箱光彩夺目却彼此孤立的事实，大多来自费马与欧拉。高斯把它们熔成一套有序的理论。其皇冠，是二次互反律——一条关于「不同钟面上的平方数」的深刻而出人意料的规则——他十八岁时发现，并在此书中第一次证明，称之为自己的「黄金定理」。日后他为它给出了八个不同的证明。

它为何重要

在《算术研究》之前，数论是天才们的业余爱好；在它之后，数论成了一门科学。高斯给了这门学科一种语言（同余与 ≡ 号）、一种方法（按关联次序展开的定义与证明），以及一批可供后人垒砌的深刻定理。一代又一代数学家——狄利克雷、黎曼、戴德金——都从这本书里学会了这门学问。据说高斯曾称数学为科学的女王，而数论是数学的女王；正是这本书，为她加了冕。

一个日常画面

想想普通的十二小时制钟。11 点过三个钟头不是 14 点，而是 2 点——你绕了一圈，又多走了两格。这恰好就是 a ≡ b (mod 12)。再问一个高斯钟爱的怪问题：哪些刻度，可以靠「平方」落到？1×1 是 1，2×2 是 4，3×3 是 9，4×4 是 16、绕回到 4……有些刻度你到得了，有些你永远到不了。那些到得了的刻度，就是「二次剩余」；而把不同钟面上的它们悄悄连起来的隐秘规则，正是高斯的黄金定理。

它在知识谱系中的位置

在本馆中，数论始于欧几里得（约公元前 300 年）：他证明了素数永不穷尽，并留下了求最大公约数的算法。此后两千年，这门学科靠零星的天才之作生长——先是费马，再是欧拉。高斯是那个枢纽，把这些结果拧成了一个系统。从他的书中流出狄利克雷的解析数论、黎曼对素数的研究、戴德金的理想。而他第三节里的钟面算术，正是现代密码学的舞台：Diffie–Hellman 密钥交换与 RSA，都跑在高斯的钟上。

The original document

Original source text

Carl Friedrich Gauss · Disquisitiones Arithmeticae · Leipzig: Gerh. Fleischer, 1801 · in Latin

The book is dedicated to Gauss's patron, Carl Wilhelm Ferdinand, Duke of Brunswick, and opens with a short preface setting its subject — the theory of integers, or “higher arithmetic” — apart from the arithmetic of everyday calculation. What follows is organised into seven sections and 366 numbered articles.

Section I · Congruent numbers in general (Art. 1–12)

If the number a measures the difference of the numbers b, c, then b and c are said to be congruent according to a; if not, incongruent; this a we call the modulus. Each of the numbers b, c are called a residue of the other in the first case, a nonresidue in the second.

This is Article 1 — the definition on which the whole book rests. Article 2 then fixes the notation we still use, choosing the sign ≡ for its likeness to equality:

Numerorum congruentiam hoc signo, ≡, in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, −16 ≡ 9 (mod. 5), −7 ≡ 15 (mod. 11).

(“We shall in future denote the congruence of numbers by this sign, ≡, attaching the modulus in parentheses where necessary.”) Every later section is built outward from this one idea.

Sections II–III · Linear congruences, and residues of powers (Art. 13–93)

Section II solves congruences of the first degree and proves the uniqueness of prime factorisation and the result now taught as the Chinese Remainder Theorem. Section III studies the powers of a number on the clock — the order of a residue, the existence of primitive roots for a prime modulus, and what is now called Fermat's little theorem.

Section IV · Congruences of the second degree — the fundamental theorem (Art. 94–152)

Here Gauss asks which numbers are perfect squares on a clock (the quadratic residues) and proves the law that governs them — the result he singled out as the theorema fundamentale and privately called his “golden theorem”. In Article 131 he states it:

If p is a prime number of the form 4n+1, then +p, but if p is of the form 4n+3, then −p, will be a residue or nonresidue of any prime number which, taken positively, is a residue or nonresidue of p.

He gave the first complete proof — Legendre had stated the law earlier, but his proof had a gap — and supplied a second, entirely different proof in the same book.

Section V · Quadratic forms (Art. 153–307)

By far the longest section: a deep theory of binary and ternary quadratic forms ax² + bxy + cy², their composition into a group-like structure, and the “genus” classification — machinery that would grow, in the next century, into algebraic number theory.

[ … ]

Section VII · The division of the circle (Art. 335–366)

The book closes with a surprise: the algebra of cutting a circle into equal parts turns out to be number theory in disguise. Gauss proves which regular polygons can be drawn with straightedge and compass — exactly those whose number of sides is a power of two times distinct Fermat primes — and in particular that the regular 17-gon is constructible, the discovery that, at eighteen, had decided him on mathematics.

Carl Friedrich Gauss · Brunswick, 1801