Translation:Disquisitiones Arithmeticae

Recherches_arithmétiques

CONTENTS


 * /Dedication
 * /Preface
 * On Congruent Numbers in General
 * Congruent numbers, moduli, residues and non-residues, 1.
 * Minimal residues, 4.
 * Elementary propositions about congruences, 5.
 * Some applications, 12.
 * 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.
 * 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 $p$, the number of terms in its period is a divisor of $p-1$ , 49.
 * Fermat's Theorem, 50.
 * How many numbers generate a period whose multitude is a given divisor of $p-1$, 52.
 * Primitive roots, bases, indices, 57.
 * Algorithm for computing indices, 58.
 * On the roots of the congruence $ 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.
 * 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 $-1$, 108.
 * The residues $+2$ and $-2$, 112.
 * The residues $+3$ and $-3$, 117.
 * The residues $+5$ and $-5$, 121.
 * The residues $+7$ and $-7$, 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.
 * 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 $\sqrt{(bb-ac)} \pmod{M}$ to which  representations of the number $M$  by the form $(a,b,c)$  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 $-1, +2, -2$, 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 $ax^2 + by^2 + cz^2=0$, 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.
 * Various Applications of the Preceding Investigations
 * Reduction of fractions into simpler fractions, 309.
 * Conversion of ordinary fractions to decimals, 312.
 * Solving the congruence $xx \equiv A$ by the method of exclusion, 319.
 * Solving the indeterminate equation $mxx+nyy=A$ by exclusions, 323.
 * Another method of solving the congruence $xx \equiv A$, in the case where $A$ is negative, 327.
 * Two methods of distinguishing composite numbers from prime numbers, and of determining their factors, 329.
 * 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 $x^n-1=0$, 337.
 * Theory of the roots of this equation (where it is assumed that $n$ is a prime number)
 * Omitting the root 1, the others will be given by the equation $X=x^{n-1}+x^{n-2}+\dots+x+1=0$ . The function $X$ cannot be decomposed into factors of lesser degree with rational coefficients, 341.
 * A plan for the following investigation is stated, 342.
 * All roots $\Omega$ are distributed in certain classes (periods), 343.
 * Various theorems on these periods, 344.
 * These investigations are applied to the solution of the equation $X=0$, 352.
 * Examples for $n=19$, in which the difficulty is reduced to two third degree equations and one second degree equation, and for $n=17$ , 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 $\Omega$ into two periods, 356.
 * Return to the demonstration of the theorem in Sect. IV, 357.
 * On the equation for distributing the root $\Omega$ into three periods, 358.
 * The equations giving the roots $\Omega$ 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 $\Omega$, 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