Sách gồm 13 chương với mục lục như sau
Chương I DIVISIBILITY AND INDETERMINATE EQUATIONS OF FIRST DEGREE
§ 1. Divisibility
§ 2. Least common multiple
§ 3. Greatest common divisor
§ 4. Relatively prime numbers
§ 5. Relation between the greatest common divisor and the least common multiple
§ 6. Fundamental theorem of arithmetic
§ 7. Proof of the formulae ...
§ 8. Rules for calculating the greatest common divisor of two numbers
§ 9. Rrepresentation of rationals as simple continued fractions
§ 10. Linear form of the greatest common divisor
§ 11.FIndeterminate equations of m variables and degree 1
§ 12. Chinese Remainder Theorem
§ 13. Thue Theorem
§ 14. Square-free numbers
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Chương II DIOPHANTINE ANALYSIS OF SECOND AND HIGHER DEGREES
§ 1. Diophantine equations of arbitrary degree and one unknown
§ 2. Problems concerning Diophantine equations of two or more unknowns
§ 3. The equation x2 + y2 = z2
§ 4 .Integral solutions of the equation ...
§ 5. Pythagorean triangles of the same area
§ 6. On squares whose sum and difference are squares
§ 7. The equation x4 + y4 = z2
§ 8. On three squares for which the sum of any two is a square
§ 9. Congruent numbers
§ 10. The equation x2 + y2 + z2 = t2
§ 11. The equation xy = zt
§ 12. The equation x4 - x2y2 + y4 = z2
§ 13. The equation x4+9x2y2 + 27y4 = z2
§ 14. The equation x3 + y3 = 2z3
§ 15. The equation x3 + y3 = az3 with a>2
§ 16. Triangular numbers
§ 17. The equation x2 Dy2 = 1
§ 18. The equations x2 + k = y3 where k is an integer
§ 19. On some exponential equations and others
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Chương III PRIME NUMBERS
§ 1. The primes. Factorization of a natural number m into primes
§ 2. The Eratosthenes sieve. Tables of prime numbers
§ 3. The differences between consecutive prime numbers
§ 4. Goldbach's conjecture
§ 5. Arithmetical progressions whose terms are prime numbers
§ 6. Primes in a given arithmetical progression
§ 7. Trinomial of Euler x2 + x + 41
§ 8. The conjecture H
§ 9. The function ...
§ 10. Proof of Bertrand's postulate (Theorem of Tchebycheff)
§ 11. Theorem of H. F. Scherk
§ 12. Theorem of H. E. Eichert
§ 13. A conjecture on prime numbers
§ 14. Inequalities for the function ...
§ 15. The prime number theorem and its consequences
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Chương IV NUMBER OF DIVISORS AND THEIR SUM
§ 1. Number of divisors
§ 2. Sums d(1) + d(2) + ... + d(n)
§ 3. Numbers d(n) as coefficients of expansions
§ 4. Sum of divisors
§ 5. Perfect numbers
§ 6. Amicable numbers
§ 7. The sum ...
§ 8. The numbers ... as coefficients of various expansions
§ 9. Sums of summands depending on the natural divisors of a natural number n
§ 10. Mobius function
§ 11. Liouville function ...
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Chương V CONGRUENCES
§ 1. Congruences and their simplest properties
§ 2. Roots of congruences. Complete set of residues
§ 3. Eoots of polynomials and roots of congruences
§ 4. Congruences of the first degree
§ 5. Wilson's theorem and the simple theorem of Fermat
§ 6. Numeri idonei
§ 7. Pseudoprime and absolutely pseudoprime numbers
§ 8. Lagrange's theorem
§ 9. Congruences of the second degree
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Chương VI EULER'S TOTIENT FUNCTION AND THE THEOREM OF EULER
§ 1. Euler's totient function
§ 2. Properties of Euler's totient function
§ 3. The theorem of Euler
§ 4. Numbers which belong to a given exponent with respect to a given modulus
§ 5. Proof of the existence of infinitely many primes in the arithmetical progression nk +1
§ 6. Proof of the existence of the primitive root of a prime number
§ 7. An nth power residue for a prime modulus p
§ 8. Indices, their properties and applications
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Chương VII REPRESENTATION OF NUMBERS BY DECIMALS IN A GIVEN SCALE
§ 1.Representation of natural numbers by decimals in a given scale
§ 2. Representations of numbers by decimals in negative scales
§ 3. Infinite fractions in a given scale
§ 4. Representations of rational numbers by decimals
§ 5. Normal numbers and absolutely normal numbers
§ 6. Decimals in the varying scale
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Chương VIII CONTINUED FRACTIONS
§ 1. Continued fractions and their convergents
§ 2. Representation of irrational numbers by continued fractions
§ 3. Law of best approximation
§ 4. Continued fractions of quadratic irrationals
§ 5. Application of the continued fraction for ...
§ 6. Continued fractions other than simple continued fractions
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Chương IX LEGENDRE'S SYMBOL AND JACOBI'S SYMBOL
§ 1. Legendre's symbol (D-p) and its properties
§ 2. The quadratic reciprocity law
§ 3. Calculation of Legendre's symbol by its properties
§ 4. Jacobi's symbol and its properties
§ 5. Eisentein's rule
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Chương X MERSENNE NUMBERS AND FERMAT NUMBERS
§ 1. Some properties of Mersenne numbers
§ 2. Theorem of E. Lucas and D. H. Lehmer
§ 3. How the greatest of the known prime numbers have been found
§ 4. Prime divisors of Fermat numbers
§ 5. A necessary and sufficient condition for a Fermat number to be a prime
§ 6. How the fact that number ...
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Chương XI REPRESENTATIONS OF NATURAL NUMBERS AS SUMS OF NON-NEGATIVE kth POWERS
§ 1. Sums of two squares
§ 2. The average number of representations as sums of two squares
§ 3. Sums of two squares of natural numbers
§ 4. Sums of three squares
§ 5. Representation by four squares
§ 6. The sums of the squares of four natural numbers
§ 7. Sums of m > 5 positive squares
§ 8. The difference of two squares
§ 9. Sums of two cubes
§ 10. The equation x3 + y3 = z3
§ 11. Sums of three cubes
§ 12. Sums of four cubes
§ 13. Equal sums of different cubes
§ 14. Sums of biquadrates
§ 15. Waring's theorem
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Chương XII SOME PROBLEMS OF THE ADDITIVE THEORY OF NUMBERS
§ 1. Partitio numerorum
§ 2. Representations as sums of n non-negative summands
§ 3. Magic squares
§ 4. Schur's theorem and its corollaries
§ 5. Odd numbers which are not of the form 2k+p, where p is a prime
Chương XIII COMPLEX INTEGERS
§ 1. Complex integers and their norm. Associated integer
§ 2. Euclidean algorithm and the greatest common divisor of complex integers
§ 3. The least common multiply of complex integers
§ 4. Complex primes
§ 5. The factorization of complex integers into complex prime factors
§ 6. The number of complex integers with a given norm
§ 7. Jacobi's four-square theorem
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