导语

  

内容提要

  

    C.F.高斯著的《算术探究(英文版)》主要由七部分组成：第一部分同余数基本介绍，第二部分一次同余式，第三部分幂的乘余，第四部分二次同余数。第五部分型和二次不定方程。第六部分是对之前讨论的各种应用介绍。第七部分定义圆截面方程。读者对象：从事理论学习的研究生和数学工作者。

Translator's Preface
Bibliographical Abbreviations
Dedication
Author's Preface
Section I. Congruent Numbers in General
  Congruent numbers, moduli, residues, and nonresidues,
    art. 1 ft.
  Least residues, art. 4
  Elementary propositions regarding congruences, art. 5
  Certain applications, art. 12
Section II. Congruences of the First Degree
  Preliminary theorems regarding prime numbers, factors, etc.,
    art. 13
  Solution of congruences of the first degree, art. 26
  The method of finding a number congruent to given residues
    relative to given moduli, art. 32
  Linear congruences with several unknowns, art. 37
  Various theorems, art. 38
Section III. Residues of Powers
  The residues of the terms of a geometric progression which
    begins with unity constitute a periodic series, art. 45
  If the modulus = p (a prime number), the number of terms in
    its period is a divisor of the number p - 1, art. 49
  Fermat's theorem, art, 50
  How many numbers correspond to a period in which the
    number of terms is a given divisor of p - 1, art. 52
  Primitive roots, bases, indices, art. 57
  Computation with indices, art. 58
  Roots of the congruence x" = A, art. 60
  Connection between indices in different systems, art. 69
    Bases adapted to special purposes, art. 72
    Method of finding primitive roots, art. 73
    Various theorems concerning periods and primitive roots, art. 75
  A theorem of Wilson, art. 76
    Moduli which are powers of prime numbers, art. 82
    Moduli which are powers of the number 2, art. 90
    Moduli composed of more than one prime number, art. 92
Section IV. Congruences of the Second Degree
  Quadratic residues and nonresidues, art. 94
  Whenever the modulus is a prime number, the number of
  residues less than the modulus is equal to the number of
  nonresidues, art. 96
    The question whether a composite number is a residue or
  nonresidue of a given prime number depends on the nature
  of the factors, art. 98
    Moduli which are composite numbers, art. 100
    A general criterion whether a given number is a residue or a
  nonresidue of a given prime number, art. 106
    The investigation of prime numbers whose residues or non-residues are given numbers, art. 107
    The residue - 1, art. 108
    The residues + 2 and - 2, art. 112
    The residues + 3 and - 3, art. 117
    The residues +5 and -5, art. 121
    The residues +7and -7, art. 124
    Preparation for the general investigation, art. 125
  By induction we support a general (fundamental) theorem
    and draw conclusions from it, art. 130
    A rigorous demonstration of the fundamental theorem,
  art. 135
    An analogous method of demonstrating the theorem of
  art. 114, art. 145
    Solution of the general problem, art. 146
  Linear forms containing all prime numbers for which a given
    number is a residue or nonresidue, art. 147
  The work of other mathematicians concerning these in-
  vestigations, art. 151
    Nonpure congruences of the second degree, art. 152
  Section V. Forms and Indeterminate Equations of the Second Degree
  Plan of our investigation ; definition of forms and their notation,
    art. 153
  Representation of a number; the determinant, art. 154
  Values of the expression (b2- ac) (mod. M) to which
    belongs a representation of the number M by the form
    (a, b, c), art. 155
  One form implying another or contained in it; proper and
    improper transformation, art. 157
  Proper and improper equivalence, art. 158
  Opposite forms, art. 159
  Neighboring forms, art. 160
  Common divisors of the coefficients of forms, art. 161
  The connection between all similar transformations of a
    given form into another given form, art. 162
  Ambiguous forms, art. 163
  Theorem concerning the case where one form is contained in
    another both properly and improperly, art. 164
  General considerations concerning representations of num-
    bers by forms and their connection with transformations,
    art. 166
  Forms with a negative determinant, art. 171
  Special applications for decomposing a number into two
    squares, into a square and twice a square, into a square
    and three times a square, art. 182
  Forms with positive nonsquare determinant, art. 183
  Forms with square determinant, art. 206
  Forms contained in other forms to which, however, they are
    not equivalent, art. 213
  Forms with 0 determinant, art. 215
  The general solution by integers of indeterminate equations
    of the second degree with two unknowns, art. 216
  Historical notes, art. 222
  Distribution of forms with a given determinant into classes,
    art. 223
  Distribution of classes into orders, art. 226
  The partition of orders into genera, art. 228
  The composition of forms, art. 234
  The composition of orders, art. 245
  The composition of genera, art. 246
  The composition of classes, art. 249
  For a given determinant there are the same number of classes
  in every genus of the same order, art. 252
  Comparison of the number of classes contained in individual
  genera of different orders, art. 253
  The number of ambiguous classes, art. 257
    Half of all the characters assignable for a given determinant
  cannot belong to any properly primitive genus, art. 261
    A second demonstration of the fundamental theorem and the
  other theorems pertaining to the residues -1, +2, -2,
  art. 262
    A further investigation of that half of the characters which
  cannot correspond to any genus, art. 263
    A special method of decomposing prime numbers into two
  squares, art. 265
  A digression containing a treatment of ternary forms,
  art. 266 ff.
    Some applications to the theory of binary forms, art. 286 IT.
  How to find a form from whose duplication we get a given
    binary form of a principal genus, art. 286
  Except for those characters for which art. 263, 264 showed it
  was impossible, all others will belong to some genus,
  art. 287
    The theory of the decomposition of numbers and binary
  forms into three squares, art. 288
    Demonstration of the theorems of Fermat which state that
  any integer can be decomposed into three triangular numbers
  or four squares, art. 293
    Solution of the equation ax2 + by2 + cz2 = 0, art. 294
    The method by which the illustrious Legendre treated the
  fundamental theorem, art. 296
    The representation of zero by ternary forms, art. 299
    General solution by rational quantities of indeterminate
  equations of the second degree in two unknowns, art. 300
    The average number of genera, art. 301
    The average number of classes, art. 302
    A special algorithm for properly primitive classes; regular
  and irregular determinants etc., art. 305
Section VI. Various Applications of the Preceding Discussions
    The resolution of fractions into simpler ones, art. 309
    The conversion of common fractions into decimals, art. 312
    Solution of the congruence x2 = A by the method of exclusion, art. 319
    Solution of the indeterminate equation mx2 + ny2 = A by
  exclusions, art. 323
    Another method of solving the congruence x2 - A for the
    case where ,4 is negative, art. 327
    Two methods for distinguishing composite numbers from
  primes and for determining their factors, art. 329
Section VII. Equations Defining Sections of a Circle
  The discussion is reduced to the simplest case in which the
    number of parts into which the circle is cut is a prime
    number, art. 336
  Equations for trigonometric functions of arcs which are a
    part or parts of the whole circumference; reduction of
    trigonometric functions to the roots of the equation
    xn - 1 = 0, art. 337
  Theory of the roots of the'equation x" - I = 0 (where n
    is assumed to be prime), art. 341 ft.
  Except for the root 1, the remaining roots contained in (Ω)
    are included in the equation X = xn-1 + xn-2 + etc.
    + x + 1 = 0; the function X cannot be decomposed into
    factors in which all the coefficients are rational, art. 341
  Declaration of the purpose of the following discussions,
   art. 342
  All the roots in (fl) are distributed into certain classes
   (periods), art. 343
  Various theorems concerning these periods, art. 344
  The solution of the equation X = 0 as evolved from the
    preceding discussions, art. 352
  Examples for n = 19 where the operation is reduced to the
    solution of two cubic and one quadratic equation, and
    for n = 17 where the operation is reduced to the solution of
    four quadratic equations, art. 353, 354
  Further discussions concerning periods of roots, art. 355 ft.
    Sums having an even number of terms are real quantities,
    art. 355
  The equation defining the distribution of the roots (Ω) into
    two periods, art. 356
  Demonstration of a theorem mentioned in Section IV,
    art. 357
  The equation for distributing the roots (Ω) into three periods,
    art. 358
  Reduction to pure equations of the equations by which the
    roots (Ω) are found, art. 359
  Application of the preceding tO trigonometric functions,
    art. 361 ft.
  Method of finding the angles corresponding to the individual
    roots of (Ω), art. 361
  Derivation of tangents, cotangents, secants, and cosecants
    from sines and cosines without division, art. 362
  Method of successively reducing the equations for trigonometric functions, art. 363
  Sections of the circle which can be effected by means of
    quadratic equations or by geometric constructions, art. 365
Additional Notes
Tables
Gauss' Handwritten Notes
List of Special Symbols
Directory of Terms