Introduction to the Theory of Numbers
Prof. Ghiocel Groza

Department of Mathematics and Informatics,
Technical University of Civil Engineering Bucharest,
Romania

Ch. 1. Divisibility

1.1. Divisibility. Properties of the greatest common divisor and the least common
multiple. The Euclidean algorithm
1.2. Primes. The fundamental theorem of arithmetic. Brief mention of some unsolved
problems concerning prime numbers

Ch. 2. Congruences

2.1 General properties of the congruences. Fermat’s theorem. Euler’s generalization of
Fermat’s theorem. Wilson’s theorem
2.2. Congruences of degree 1. Chinese remainder theorem. Properties of Euler’s ?-
function
2.3. Congruences of higher degree. Prime power moduli. Primes Modulus. Power
residues
2.4. Number theory from an algebraic viewpoint

Ch. 3. Quadratic Reciprocity

3.1. Quadratic residues. Legendre symbol. Lemma of Gauss
3.2. The Gaussian reciprocity law. Jacobi symbol

Ch. 4. Some Functions of Number Theory

4.1. Greatest integer function
4.2. Numerical functions
4.3. Moebius function. The Moebius inversion formula
4.4. Recurrence functions. The Fibonacci numbers

Ch. 5. Some Diophantine Equations

5.1. The equation ax+by=c
5.2. The equations and
5.3. Sum of four squares
5.4. Warning’s problem. Sum of fourth powers
5.5. Sum of two squares
5.6.The equation
5.7. The equation
5.8. Binary quadratic forms. Equivalence of quadratic forms

Ch. 6. Farey Fractions

6.1. Farey sequences
6.2. Rational approximations

Ch. 7. Simple Continued Fractions

7.1. The continued fraction expansion of a rational number. Finite continued fraction
7.2. Infinite simple continued fractions. Expansion of an irrational number.
Approximations to irrational numbers. Best possible approximations
7.3. Periodic continued fractions
7.4. Pell’s equation

Ch. 8. Elementary Remarks on the Distribution of Primes

8.1. The function
8.2. The sequence of primes
8.3. Bertrand’s postulate

Ch. 9. Algebraic Numbers

9.1. Polynomials with rational coefficients. Primitive polynomials. Gauss’ Lemma
9.2. Algebraic numbers. Algebraic integers
9.3. Algebraic number fields and their ring of the algebraic integers

References

1. T. Apostol, Introduction to Analytic Number Theory, Springer Verlag, New York,
1976.
2. I. Niven and H. Zuckerman, An Introduction to the Theory of Numbers, John Wiley &
Sons, New York, 1966.