Elementary Number Theory

Transcription

1 Elementary Number Theory

2 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What does the Elementary in the title refer to? The treatment is NOT based on notions and results from other branches of mathematics, e.g. algebra and/or analysis. Notation N denotes the set of positive integers {1, 2, 3,...} The set of all integers is Z = {0, ±1, ±2,...}.

3 History and overview History The study of integers has its origins in China and India, e.g. Chinese Remainder Theorem, around 1000 BC. A systematic treatment of these questions started around 300 BC in Greek. Basic Problem The basic problem in the theory of numbers is to decide if a given integer N has a factorization, N = pq for integers p, q, the so-called divisors of N. Primes If N has only the trivial divisors 1 and N, we say N is prime. Here are the first 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

4 History and overview Largest Prime Largest known prime: 2 57,885,161 1, a number with 17, 425, 170 digits and was discovered in 2013, and it is the 48th known Mersenne prime. Euclid (300 BC) More than 2000 years ago Euclid proved that there are infinitely many primes. every number N has a factorization into prime numbers: N = p 1 p n, for not necessarily distinct prime numbers. Euclidean algorithm for the greates common divisor.

5 History and overview Gauss ( ) About 2000 years later Gauss was the first to prove the uniqueness of this factorization up to order of factors. The statement goes by the name of Fundamental Theorem of Arithmetic. In 1801 Gauss introduced in his Disquintiones Arithmaticae, the first modern book on number theory, the theory of congruences: Two integers a and b are congruent modulo m, if m divdes a b. We denote this by a b mod m. We will devote a substantial part on the theory of congruences, because it allows one to carry out addition, multiplication and exponentiation modulo m much faster than in Z.

6 History and overview Diophantus of Alexandria Another Greek mathematician, Diophantus of Alexandria, initated the study of the solutions of polynomial equations in two or more variables in the integers. The modern developments that grew out of this basic quest, is known as diophantine equations and there is also a mathematical branch, called Diophantine Geometry. The most famous diophantine equation appears in Fermat s Last Theorem: There are no non-zero integers x, y, z such that x n + y n = z n for any n 3. In 1995 Andrew Wiles proved this result using the theory of elliptic curves. The search for a proof of Fermat s Last Theorem led to many discoveries in mathematics.

7 History and overview Heros of our course Pierre de Fermat (1601/7-1665): Fermat s Little Theorem, factorization of integers,... Leonhard Euler ( ): Euler s totient function, extension of Fermat s Little Theorem,... Carl Friedrich Gauss ( ): Congruences, Quadratic Reciprocity Theorem,...

8 Fundamental Problems The big quest in number theory is to factor large numbers. The naive trial division by 2 and all odd integers less than N does not provide a fast method. Although many algorithms have been developed to deal with this fundamental problem, there still is no fast factorization algorithm. In mathematics, if you are not able to settle a problem, you are trying to find a variation that is more feasible. In our case, we are interested in if a given number N is prime or composite, i.e. are there primality testing algorihms. In 2004 Agarwal, Kayal and Saxena proved that there exists a good primality testing algorithm.

9 Fundamental Problems Another fundamental problem in number theory is to understand the nature of prime numbers. Green-Tao (2004): There exist arbitraliy long arithmetic progressions of prime numbers. Zhang (2013): Prove of the bounded gap conjecture for primes. There are infinitely many pairs of primes that differ by at most 70, 000, 000. In other words, that the gap between one prime and the next is bounded by 70, 000, 000 infinitely often. Tao and his collaborators in Polymath 8 were able to reduce the gap from 70 millions down to The ultimate goal is to get down to 2, this is known as the twin-prime conjecture.

10 Fundamental Problems and Applications Riemann connected the distribution of primes with the zeros of a certain function, the zeta function, and conjectured that all the non-trivial zeros lie on the critical line. If the conjecture is true, then the distribution of primes ialerts optimal in some sense. A solution of this conjecture would earn you US Dollars from the Clay Institute! There is a second Clay Millenium problem about number theory: Birch and Swinnerton-Dyer conjecture. Cryptosystems Technology has added an algorithmic side to number theory and provides a lot of tools to experiment with numbers and search for hidden properties. Finally, elementary number theory makes a secure transfer of information possible!

11 Fundamental Problems and Applications RSA In this course we will discuss the RSA-algorithm due to Rivest, Shamir and Adleman from RSA-challenges: RSA-2048 asks you to factorize a number with 2048 binary digits and as reward offers US Dollars.

12 Euclid

13 Pierre de Fermat

14 Blaise Pascal

15 Leonhard Euler

16 Carl Friedrich Gauss

17 Bernhard Riemann

18 Godfrey H. Hardy

19 Leonhard Euler

20 Srinivasa Ramanujan

21 Paul Erdoes

22 Atle Selberg

23 Andre Weil

24 John Tate

25 Pierre Deligne

26 Ben Green

27 Terence Tao

28 Yitang Zhang