Course

Unit I

Divisibility: Definition, properties, division algorithm, greatest integer function (Sec 1.1)

Primes: Definition, Euclid’s Theorem, Prime Number Theorem (statement only), Goldbach and Twin Primes conjectures, Fermat primes, Mersenne primes. The greatest common divisor: Definition, properties, Euclid’s algorithm, linear combinations and the GCD – The least common multiple: Definition and properties. The Fundamental Theorem of Arithmetic: Euclid’s Lemma, canonical prime factorization, divisibility, gcd, and lcm in terms of prime factorizations. Primes in arithmetic progressions: Dirichlet’s Theorem on primes in arithmetic progressions (statement only) (Sec 1.2 to 1.5)

Unit II

Congruences: Definitions and basic properties, residue classes, complete residue systems, reduced residue systems – Linear congruences in one variable, Euclid’s algorithm – Simultaneous linear congruences, Chinese Remainder Theorem – Wilson’s Theorem – Fermat’s Theorem, pseudoprimes and Carmichael numbers – Euler’s Theorem (Sec 2.1 to 2.6).

Unit III

Arithmetic functions: Arithmetic function, multiplicative functions: definitions and basic examples – The Moebius function, Moebius inversion formula – The Euler phi function, Carmichael conjecture – The number-of-divisors and sum-of-divisors functions – Perfect numbers, characterization of even perfect numbers (Sec 3.1 to 3.6).

Unit IV

Quadratic residues: Quadratic residues and nonresidues – The Legendre symbol: Definition and basic properties, Euler’s Criterion, Gauss’ Lemma – The law of quadratic reciprocity (Sec 4.1 to 4.3).

Unit V

Primitive roots:

The order of an integer – Primitive roots: Definition and properties – The Primitive Root Theorem: Characterization of integers for which a primitive root exists (Sec 5.1 to 5.3).

Diophantine Equations

Linear Diophantine Equations – Pythagorean triples – Representation of an integer as a Sum of squares (Sec 6.1, 6.3, 6.5).

Objectives: To enable students to

• Understand the concept of divisibility, congruencies and arithmetical funcitons
• Understand the concept of primitive roots and Diophantine equations

Text book:

James Strayer, ‘Elementary Number Theory’, Waveland Press, 1994/2002, ISBN 1-57766-224-5 

References:

1. Tom M. Apostol,’ Introduction to Analytic Number Theory’, Springer, Under Graduate Studies in Mathematics, 1976.
2. Kenneth Rosen, Elementary Number Theory and its Applications, 5th Edition, McGraw Hill.
3. Niven, H. Zuckerman, H. Montgomery, An Introduction to the Theory of Numbers, 5th Edition, Wiley.
4. Burton, David M. Elementary Number Theory. Allyn and Bacon, 1976.