Yau Tsinghua Mathcamp
All truths are easy to understand once they are discovered,
the point is to discover them.
- Galileo.

# Yau Tsinghua Mathcamp

## Number Theory

Instructor: Philip Vetter (Institute of Deep-Sea Science and Engineering, CAS)

Coaches: Teng Wang 王腾 (USTC)、 Han Wu 吴晗 (Hubei U)

Location: A103

Course description:

Number theory studies the properties and patterns of the integers and their relationships.

This field has ancient beginnings, dating back at least to the ancient Babylonians, who 1700 years before Christ, had an algorithm for creating Pythagorean triplets $\inline&space;&space;(a,b,c)$ with $\inline&space;a^2&space;+&space;b^2&space;=&space;c^2$. The classical Greeks studied the properties of different kinds of numbers; Euclid organized and formalized many patterns and properties.

Number theory has major modern advances; the proof of Fermat's last theorem happened in 1993, and proofs about gaps in prime numbers were created even more recently. Today number theory has a practical application in cryptographic systems which secure internet communications and transactions.

This course aims to develop problem-solving skills and provide a foundation for further studies in advanced number theory.

In this course we will cover topics such as:

• Historical overview and relevance of number theory
• Divisibility, congruences and modular arithmetic
• Euclidean algorithm and greatest common divisor
• Prime numbers, prime factorization and the fundamental theorem of arithmetic
• Diophantine Equations, Pythagorean triples and Fermat's Last Theorem
• Number-theoretic functions, multiplicative functions and their properties
• Quadratic residues and non-residues, law of quadratic reciprocity
• Quadratic forms and applications
• Sieve of Eratosthenes and primality testing
• Distribution of prime numbers.
• Prime number gaps and prime number patterns
• Introduction to number-theoretic cryptography