數學 1801

《算術研究》

卡爾·弗里德里希·高斯

他把算術變成一面鐘，奠定了現代數論。

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In depth · the introduction

卡爾·弗里德里希·高斯拿起整數這個雜亂的世界，發現它其實像一面鐘一樣運轉。

核心思想

在鐘面上，9 + 5 不是 14——而是 2。過了 12，就繞回起點。高斯把這種「繞回」變得既精確又普適。任取一個數 n，叫它「模數」，此後兩個數若除以 n 留下相同的餘數，就算「一樣」。他寫作 a ≡ b (mod n)，並把 n 稱為模數——這個記號，乃至「模數」這個詞，我們今天仍在用。

聽起來不起眼，卻改變了一切。一個關於無窮多個整數的問題，化成了一個只有 n 個刻度的有限鐘面上的問題，而原本看不見的規律忽然就顯形了。高斯整本書，都從這一招搭起：在鐘上做算術。

它是怎麼來的

高斯出身於布倫瑞克一個貧寒之家，很早就被發現天賦，並得到當地公爵的資助。十八歲那年，他證明了兩千年來無人做到的事：僅用直尺和圓規，就能畫出正十七邊形——這件事讓他最終選擇了數學，而非語言學。他私下記著一本發現日記，二十一歲時便寫成了這部書。

在他之前，數論是一箱光彩奪目卻彼此孤立的事實，大多來自費馬與歐拉。高斯把它們熔成一套有序的理論。其皇冠，是二次互反律——一條關於「不同鐘面上的平方數」的深刻而出人意料的規則——他十八歲時發現，並在此書中第一次證明，稱之為自己的「黃金定理」。日後他為它給出了八個不同的證明。

它為何重要

在《算術研究》之前，數論是天才們的業餘愛好；在它之後，數論成了一門科學。高斯給了這門學科一種語言（同餘與 ≡ 號）、一種方法（按關聯次序展開的定義與證明），以及一批可供後人壘砌的深刻定理。一代又一代數學家——狄利克雷、黎曼、戴德金——都從這本書裡學會了這門學問。據說高斯曾稱數學為科學的女王，而數論是數學的女王；正是這本書，為她加了冕。

一個日常畫面

想想普通的十二小時制鐘。11 點過三個鐘頭不是 14 點，而是 2 點——你繞了一圈，又多走了兩格。這恰好就是 a ≡ b (mod 12)。再問一個高斯鍾愛的怪問題：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、繞回到 4……有些刻度你到得了，有些你永遠到不了。那些到得了的刻度，就是「二次剩餘」；而把不同鐘面上的它們悄悄連起來的隱祕規則，正是高斯的黃金定理。

它在知識譜系中的位置

在本館中，數論始於歐幾里得（約公元前 300 年）：他證明了質數永不窮盡，並留下了求最大公因數的演算法。此後兩千年，這門學科靠零星的天才之作生長——先是費馬，再是歐拉。高斯是那個樞紐，把這些結果擰成了一個系統。從他的書中流出狄利克雷的解析數論、黎曼對質數的研究、戴德金的理想。而他第三節裡的鐘面算術，正是現代密碼學的舞台：Diffie–Hellman 金鑰交換與 RSA，都跑在高斯的鐘上。

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