Elementary Number Theory
Franz Luef
21.8.2013
Franz Luef
MA1301
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, ...}.
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 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
Franz Luef
MA1301
History and overview
Largest Prime
Largest known prime: 257,885,161 − 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 = 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 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.
Franz Luef
MA1301
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
xn + yn = zn
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.
Franz Luef
MA1301
History and overview
Heros of our course
Pierre de Fermat (1601/7-1665): Fermat’s Little
Theorem, 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 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.
Franz Luef
MA1301
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 4682. The
ultimate goal is to get down to 2, this is known as the
twin-prime conjecture.
Franz Luef
MA1301
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 1000000 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!
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 with
2048 binary digits and as reward offers 200000 US Dollars.
Franz Luef
MA1301
Photographic Memory
Euclid
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MA1301
Photographic Memory
Pierre de Fermat
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MA1301
Photographic Memory
Blaise Pascal
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MA1301
Photographic Memory
Leonhard Euler
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MA1301
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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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MA1301
Photographic Memory
Srinivasa Ramanujan
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Photographic Memory
Paul Erdoes
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Photographic Memory
Atle Selberg
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MA1301
Photographic Memory
Andre Weil
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MA1301
Photographic Memory
John Tate
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MA1301
Photographic Memory
Pierre Deligne
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MA1301
Photographic Memory
Ben Green
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MA1301
Photographic Memory
Terence Tao
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MA1301
Photographic Memory
Yitang Zhang
Franz Luef
MA1301