Elementary Number Theory

Franz Luef

21.8.2013

Franz Luef MA1301

Overview

The course discusses properties of numbers, the most basicmathematical 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 otherbranches 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, ...}.

Franz Luef MA1301

History and overview

History

The study of integers has its origins in China and India, e.g.Chinese Remainder Theorem, around 1000 BC. A systematictreatment of these questions started around 300 BC in Greek.

Basic Problem

The basic problem in the theory of numbers is to decide if agiven 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

Largest known prime: 257,885,161 − 1, a number with17, 425, 170 digits and was discovered in 2013, and it is the48th known Mersenne prime.

Franz Luef MA1301

History and overview

Largest Prime

Largest known prime: 257,885,161 − 1, a number with17, 425, 170 digits and was discovered in 2013, and it is the48th known Mersenne prime.

Euclid (300 BC)

More than 2000 years ago Euclid proved that there areinfinitely many primes.

every number N has a factorization into prime numbers:

N = p1 · · · pn,

for not necessarily distinct prime numbers.

Euclidean algorithm for the greates common divisor.

Franz Luef MA1301

History and overview

Gauss (1777–1855)

About 2000 years later Gauss was the first to prove theuniqueness of this factorization up to order of factors. Thestatement goes by the name of Fundamental Theorem ofArithmetic.

In 1801 Gauss introduced in his DisquintionesArithmaticae, the first modern book on number theory,the theory of congruences:Two integers a and b are congruent modulo m, if mdivdes a− b. We denote this by a ≡ b mod m.We will devote a substantial part on the theory ofcongruences, because it allows one to carry out addition,multiplication and exponentiation modulo m much fasterthan in Z.

Franz Luef MA1301

History and overview

Diophantus of Alexandria

Another Greek mathematician, Diophantus of Alexandria,initated the study of the solutions of polynomial equations intwo or more variables in the integers. The moderndevelopments that grew out of this basic quest, is known asdiophantine equations and there is also a mathematicalbranch, called Diophantine Geometry.The most famous diophantine equation appears in Fermat’sLast Theorem: There are no non-zero integers x , y , z suchthat

xn + yn = zn

for any n ≥ 3. In 1995 Andrew Wiles proved this result usingthe theory of elliptic curves. The search for a proof of Fermat’sLast Theorem led to many discoveries in mathematics.

Norwegian Number Theorists

Many outstanding number theorists are from Norway: ViggoBrun, Atle Selberg (Fields medal 1950), Ernst Selmer, AxelThue

Franz Luef MA1301

History and overview

Heros of our course

Pierre de Fermat (1601/7-1665): Fermat’s LittleTheorem, factorization of integers,...

Leonhard Euler (1707-1783): Euler’s totient function,extension of Fermat’s Little Theorem,...

Carl Friedrich Gauss (1776-1855): Congruences,Quadratic Reciprocity Theorem,...

Franz Luef MA1301

Fundamental Problems

The big quest in number theory is to factor largenumbers.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 thisfundamental problem, there still is no “fast” factorizationalgorithm.In mathematics, if you are not able to settle a problem,you are trying to find a variation that is more feasible. Inour case, we are interested in if a given number N is primeor composite, i.e. are there primality testing algorihms. In2004 Agarwal, Kayal and Saxena proved that thereexists a “good” primality testing algorithm.

Franz Luef MA1301

Fundamental Problems

Another fundamental problem in number theory is tounderstand the nature of prime numbers.

Green-Tao (2004): There exist arbitraliy long arithmeticprogressions of prime numbers.

Zhang (2013): Prove of the bounded gap conjecture forprimes. There are infinitely many pairs of primes thatdiffer by at most 70, 000, 000. In other words, that the gapbetween one prime and the next is bounded by70, 000, 000 infinitely often.

Tao and his collaborators in Polymath 8 were able toreduce the gap from 70 millions down to 4682. Theultimate goal is to get down to 2, this is known as thetwin-prime conjecture.

Franz Luef MA1301

Fundamental Problems and Applications

Riemann connected the distribution of primes with thezeros of a certain function, the zeta function, andconjectured that all the non-trivial zeros lie on the criticalline. If the conjecture is true, then the distribution ofprimes ialerts optimal in some sense. A solution of thisconjecture would earn you 1000000 US Dollars from theClay Institute!There is a second Clay Millenium problem about numbertheory: Birch and Swinnerton-Dyer conjecture.

Cryptosystems

Technology has added an algorithmic side to number theoryand provides a lot of tools to experiment with numbers andsearch for hidden properties. Finally, elementary number theorymakes a secure transfer of information possible!

Franz Luef MA1301

Fundamental Problems and Applications

RSA

In this course we will discuss the RSA-algorithm due to Rivest,Shamir and Adleman from 1977.

RSA-challenges: RSA-2048 asks you to factorize a number with2048 binary digits and as reward offers 200000 US Dollars.

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Photographic Memory

Euclid

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Pierre de Fermat

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Blaise Pascal

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Leonhard Euler

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Carl Friedrich Gauss

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Bernhard Riemann

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Godfrey H. Hardy

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Leonhard Euler

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Srinivasa Ramanujan

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Paul Erdoes

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Atle Selberg

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Andre Weil

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John Tate

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Pierre Deligne

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Ben Green

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Terence Tao

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Yitang Zhang

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