Translation:Disquisitiones Arithmeticae

by Carl Friedrich Gauss

CONTENTS

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Dedication

Preface

First Section: On Congruent Numbers in General

Congruent numbers, moduli, residues and non-residues, 1.

Minimal residues, 4.

Elementary propositions about congruences, 5.

Some applications, 12.

Second Section: On Congruences of the First Degree

Preliminary theorems about prime numbers, factorizations, etc., 13.

Solution of congruences of the first degree, 26.

On finding numbers congruent to given residues with respect to given moduli, 32.

Linear congruences involving several unknowns, 37.

Various theorems, 38.

Third Section: On Residues of Powers

The residues of the terms of a geometric progression starting from unity constitute a periodic series, 45.

On moduli which are prime numbers

Given a prime modulus ${\textstyle p}$, the number of terms in its period is a divisor of ${\textstyle p-1}$, 49.

Fermat's Theorem, 50.

How many numbers generate a period whose multitude is a given divisor of ${\textstyle p-1}$, 52.

Primitive roots, bases, indices, 57.

Algorithm for computing indices, 58.

On the roots of the congruence ${\textstyle x^{n}\equiv A}$, 60.

Relationships between indices in different systems, 69.

Bases adapted to particular purposes, 72.

Method for finding primitive roots, 73

Various theorems on periods and primitive roots, 75

Wilson's Theorem, 76

On moduli which are prime powers, 82.

On moduli which are powers of two, 90.

On moduli which are composed of several prime factors, 92.

Fourth Section: On Congruences of the Second Degree

Quadratic residues and non-residues, 94.

When the modulus is a prime number, the number of quadratic residues is equal to the number of non-residues, 96.

The question of whether a composite number is a quadratic residue or non-residue modulo a given prime number depends on the nature of its factors, 98.

On composite moduli, 100.

General criterion by which one can determine whether a given number is a quadratic residue or non-residue modulo a given prime number, 106.

Investigations of prime numbers for which given numbers are quadratic residues or non-residues, 107.

The residue ${\textstyle -1}$, 108.

The residues ${\textstyle +2}$ and ${\textstyle -2}$, 112.

The residues ${\textstyle +3}$ and ${\textstyle -3}$, 117.

The residues ${\textstyle +5}$ and ${\textstyle -5}$, 121.

The residues ${\textstyle +7}$ and ${\textstyle -7}$, 124.

Preparation for a general investigation, 125.

A general (fundamental) theorem is established by induction; conclusions are deduced from it, 130.

Rigorous demonstration of this theorem, 135.

Analogous method to demonstrate the theorem of Article 114, 145.

Solution of the general problem, 146.

On linear forms that contain all prime numbers for which a given number is a quadratic residue or non-residue, 147.

Work by others on this subject, 151.

On impure congruences of the second degree, 152.

Fifth Section: On Forms and Equations of the Second Degree

Plan of the investigation; definition and notation for forms, 153.

Representations of numbers; determinants, 154.

Values of the expression ${\textstyle {\sqrt {(bb-ac)}}{\pmod {M}}}$ to which representations of the number ${\textstyle M}$ by the form ${\textstyle (a,b,c)}$ belong, 155.

Forms that contain or are contained in another; proper and improper transformations, 157.

Proper and improper equivalence, 158.

Opposite forms, 159, Adjacent forms, 160.

Common divisors of coefficients of forms, 161.

Relationships between all similar transformations of one form into another, 162.

Ambiguous forms, 163.

Theorem on the case in which a form is both properly and improperly contained within another form, 164.

General considerations on the representations of numbers by forms, and their connection with transformations, 166.

Forms of negative determinant, 171

Special applications to the decomposition of numbers into two squares, into a single and double square, and into a single and triple square, 182.

On forms with positive non-square determinant, 183.

On forms with square determinant, 206.

Forms that are contained in others, but not equivalent to them, 213.

Forms with determinant 0, 215.

General integer solutions of all indeterminate equations of the second degree involving two variables, 216.

Historical remarks, 222.

Further Investigations on Forms.

Distribution into classes of forms with a given determinant, 226.

Distribution of classes into orders, 226.

Distributions of orders into genera, 228.

Composition of forms, 234.

Composition of orders, 245, genera, 246, classes, 249.

For a given determinant, each genus of the same order contains the same number of classes, 252.

Comparison of the number of classes contained in the different orders within a fixed genus, 253.

On the number of ambiguous classes, 257.

For a given determinant, half of the characters do not belong to any properly primitive (positive for negative determinant) genus, 261.

Second proof of the fundamental theorem, and of the remaining theorems regarding the residues ${\textstyle -1,+2,-2}$, 262.

The half of the characters which do not correspond to any genus, determined more precisely, 263.

Special method for decomposing a given prime as a sum of two squares, 265.

Digression containing a treatise on ternary forms.

Some applications to the theory of binary forms.

On finding a form whose duplication results in a given binary form, 286.

The genera corresponding to all characters, except for those which have been shown to be impossible in Articles 262 and 263, 287.

The theory of decomposing numbers and binary forms into three squares, 288.

Demonstration of Fermat's theorems, that every integer can be decomposed into three triangular numbers or into four squares, 293.

Solution of the equation ${\textstyle ax^{2}+by^{2}+cz^{2}=0}$, 294.

On the method by which Legendre treated the fundamental theorem, 296.

Representation of zero by arbitrary ternary forms, 299.

General solution in rational numbers of indeterminate equations of the second degree with two unknowns, 300.

On the average number of genera, 301, classes, 302.

Special algorithm for properly primitive classes; regular and irregular determinants, 305.

Sixth Section: Various Applications of the Preceding Investigations

Reduction of fractions into simpler fractions, 309.

Conversion of ordinary fractions to decimals, 312.

Solving the congruence ${\textstyle xx\equiv A}$ by the method of exclusion, 319.

Solving the indeterminate equation ${\textstyle mxx+nyy=A}$ by exclusions, 323.

Another method of solving the congruence ${\textstyle xx\equiv A}$, in the case where ${\textstyle A}$ is negative, 327.

Two methods of distinguishing composite numbers from prime numbers, and of determining their factors, 329.

Seventh Section: On the Equations Defining Divisions of the Circle

The investigation is reduced to the simplest case, in which the number of parts in which the circle is to be divide is a prime number, 336.

Equations for the trigonometric functions of arcs which consist of one or more parts of the circumference; Reduction of trigonometric functions to the roots of the equation ${\textstyle x^{n}-1=0}$, 337.

Theory of the roots of this equation (where it is assumed that ${\textstyle n}$ is a prime number)

Omitting the root 1, the others will be given by the equation ${\textstyle X=x^{n-1}+x^{n-2}+\dots +x+1=0}$. The function ${\textstyle X}$ cannot be decomposed into factors of lesser degree with rational coefficients, 341.

A plan for the following investigation is stated, 342.

All roots ${\textstyle \Omega }$ are distributed in certain classes (periods), 343.

Various theorems on these periods, 344.

These investigations are applied to the solution of the equation ${\textstyle X=0}$, 352.

Examples for ${\textstyle n=19}$, in which the difficulty is reduced to two third degree equations and one second degree equation, and for ${\textstyle n=17}$, in which it is reduced to four second degree equations, 353, 354.

Further investigations on this subject.

Periods for which the number of terms is even, always sum to real values, 355.

On the equation defining the distribution of the root ${\textstyle \Omega }$ into two periods, 356.

Return to the demonstration of the theorem in Sect. IV, 357.

On the equation for distributing the root ${\textstyle \Omega }$ into three periods, 358.

The equations giving the roots ${\textstyle \Omega }$ can always be reduced to pure ones, 359.

Application of the preceding investigations to trigonometric functions.

Method to distinguish the angles which correspond to the different roots ${\textstyle \Omega }$, 361.

Tangents, cotangents, secants, and cosecants are derived from sines and cosines, without using division, 362.

Method to successively reduce the equations for trigonometric functions, 363.

Divisions of the circle which can be performed using only quadratic equations, or equivalently, by geometric constructions, 365.

Addenda

Tables